2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationthe path integral, Wick rotation

The path integral and Euclideanisation — make time imaginary, and field theory becomes a problem of probability in four dimensions

In 2-07 we summed $e^{iS}$ over the paths $x(t)$ of one particle. In field theory, instead of paths, we sum over all functions on spacetime $\varphi(t,\vec x)$, all histories of the field. The correlation in the vacuum $\Omega$ ($\mathrm T$ is the convention of putting later times to the left) is formally written

Even for one particle $e^{iS}\mathcal Dx$ was not a measure (2-07 §03). For fields there are also infinitely many variables at each time (1-09).

Example. Cut into a lattice, (1) is a finite multiple integral. A lattice of space $32^3$ and time 64 has 2,097,152 points, and the Yang–Mills field of $SU(3)$ has 4 links per point and 8 real numbers per link, so it is a 67,108,864-fold integral Computed. The difficulties are two: the oscillation of $e^{iS}$, and the limit of finer steps and a larger box. This chapter removes the first (the second is Parts 5 and 6).

The substitution $t=-i\tau$ has appeared twice. In 2-04 §09 we only substituted it into the wave equation and saw oscillation turn into decay. In 2-07 §05 we rotated the weight of the one-particle path integral. Here we rotate the action and correlations of fields. Putting $t=-i\tau$ ($\hbar=c=1$, signs $(-,+,+,+)$, $x_4=\tau$) into the scalar field action ((7) of 2-04 with the potential energy $V$ added), from $dt=-i\,d\tau$ and $(\partial_t\varphi)^2=-(\partial_\tau\varphi)^2$,

follows. The sign of the time derivative has come into line with the space derivatives, and the four directions are equal. $S_E$ is called the Euclidean action, and if $V$ is bounded below, the weight $e^{-S_E}$ is positive and bounded. (1) is replaced by the Schwinger functions (the name is said to come from Schwinger 1958).

Example. If $V=0$, (3) is a Gaussian integral with covariance $(-\Delta+m^2)^{-1}$ ($\Delta$ is the Laplacian $\sum_\mu\partial_\mu^2$; not the $\Delta$ of the mass gap), and $S_2(x,y)=G(x-y)$ is the propagator of (10) of 1-11. Note. (1) is not a convergent integral, so putting in $t=-i\tau$ is not a change of variables but a rewriting rule. The precise content is the analytic continuation of correlation functions; seen in a single integral, it takes the form of the next section.

The two-point function of a free field of mass $m$, written in momentum, is

$\varepsilon\to+0$ fixes on which side the poles are avoided (Feynman's prescription). We first look at the component with time difference $t=0$. Then in $e^{ip\cdot x}$ the factor $e^{-ip^0t}$ from $p^0$ is 1. Setting $\omega=\sqrt{|\vec p|^2+m^2}$, the integrand in $p^0$ is $i/((p^0)^2-\omega^2+i\varepsilon)$, and the poles $\pm\sqrt{\omega^2-i\varepsilon}$ are in the fourth and second quadrants.

Here we use just one fact about complex integrals. A pole is a point where the denominator is 0. In the complex plane, integrating along a closed path that does not enclose a pole gives 0 (Cauchy's integral theorem; precisely, the condition is that the function be complex-differentiable on and inside the path, which a rational function whose denominator does not vanish satisfies) Known. So moving the contour continuously without crossing a pole does not change the value.

Turn the contour anticlockwise as $p^0=e^{i\theta}s$ ($s$ real). While $\theta$ is between 0 and 90°, the moving line sweeps only the first and third quadrants and does not cross a pole. Joining the two lines before and after turning by distant arcs of radius $R$ makes a closed path that encloses no pole. The length of the arcs is proportional to $R$ and the integrand decreases as $1/R^2$, so the contribution of the arcs vanishes as $1/R$. So the integral along the real axis equals the integral along the turned line, $\theta=90^\circ$, that is, at $p^0=ip_4$,

The integrand on the right-hand side is positive and smooth. In four dimensions $-i\,d^4p/(p^2+m^2-i\varepsilon)$ is replaced by $d^4p_E/(p_E^2+m^2)$, and this is the Euclidean propagator Known. $1/(2\omega)$ is the $\langle x^2\rangle$ of the ground state of an oscillator of frequency $\omega$ (2-06), and each Fourier component of a free field was an oscillator (2-06 §08).

The $t\ne0$ component carries the factor $e^{-ip^0t}$. If $p^0=e^{i\theta}s$ is turned into the first quadrant, on the side of $st>0$, $|e^{-ip^0t}|=e^{st\sin\theta}$ grows far away, so the contribution of the arcs does not vanish. So $t$ is also turned by the same angle in the opposite direction, as $t=e^{-i\theta}\tau$. Then at every angle along the way $p^0t=s\tau$ is real, the factor stays an oscillation of modulus 1, and the argument with arcs goes through. At $\theta=90^\circ$, $t=-i\tau$, $p^0=ip_4$, $p^0t=p_4\tau$. This is why time turns clockwise and energy anticlockwise.

Turning the other way crosses a pole and changes the sign (Figure 1, Table 1), so the direction is fixed by $i\varepsilon$. The name is said to come from Wick's (1954) treatment of the Bethe–Salpeter equation.

ComputedThe numbers in this figure were computed on this machine (trapezoidal rule with $s=\tan u$).

The value is the same whatever the angle; turning the other way past the pole only reverses the sign.

The notation follows the $\hat A=gA$ of 3-03 §01, written without hats. That is, $g$ is absorbed into $A$, $A_\mu$ are Hermitian matrices, $D_\mu=\partial_\mu-iA_\mu$, $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu-i[A_\mu,A_\nu]$ (the same as (9) of 2-04), $\mathrm{tr}(T^aT^b)=\frac12\delta^{ab}$.

The gauge field is a 1-form (1-07), so it is rotated as $A_0\,dt=A_4\,d\tau$, that is, $A_0=iA_4$. Together with $\partial_0=i\partial_4$,

In words, only the components with one time index (the components of the electric field) get one $i$, which squares to $-1$. In the Minkowski metric the square of the electric field carried $-1$, so the two cancel, and all four indices line up with the same sign. This is the same thing as the sign of $(\partial_t\varphi)^2$ coming into line with the space derivatives in (2) for the scalar field. Putting this into $S_M=-\frac1{2g^2}\int dt\,d^3x\,\mathrm{tr}\,F_{\mu\nu}F^{\mu\nu}$ ((1) of 3-03 written in hatted notation) leaves $i\cdot(-i)=1$, and with $iS_M=-S_E$,

For Hermitian $X$, $\mathrm{tr}\,X^2=\sum|X_{ij}|^2\ge0$, so $S_E\ge0$, with equality only when $F=0$. For configurations of topological charge $Q$, $S_E\ge8\pi^2|Q|/g^2$ ((12) of 3-03, 3-05) Known. Example. For configurations with $A_4=0$ and $A_j$ independent of $\tau$, $E_E=0$, so over imaginary time of length $T$, $S_E$ is $T\mathcal E$ ($\mathcal E=\frac1{g^2}\int\mathrm{tr}\,B^2d^3x$ is the magnetic energy), and the weight is $e^{-T\mathcal E}$.

The θ term stays imaginary

The topological term $\theta Q$ is different. Each component contains exactly one $F$ with a time index, so the $i$ from the rotation and the $-i$ of $dt$ cancel, and the weight is $e^{-S_E}e^{i\theta Q}$ (the sign depends on the orientation convention), which for $\theta\ne0$ is complex. Vicari–Panagopoulos (2009) write 'the complex nature of the θ term in the Euclidean QCD Lagrangian prohibits a direct MC simulation at θ ≠ 0' Known. From here on $\theta=0$.

Putting $t=-i\tau$ and an imaginary rapidity $\phi=i\alpha$ into the boost $t'=t\cosh\phi-x\sinh\phi$, $x'=x\cosh\phi-t\sinh\phi$ of 2-04, from $\cosh i\alpha=\cos\alpha$ and $\sinh i\alpha=i\sin\alpha$,

which is a rotation of the $(\tau,x)$ plane. The Lorentz group $SO(3,1)$ is replaced by the rotation group $SO(4)$ Known. The interval becomes $\tau^2+|\vec x|^2\ge0$, and the light cone and the distinction between past and future disappear. Which direction to call time is chosen afterwards. If reflection positivity (4-10) also holds, then by rotation invariance the same theory comes back whichever direction is chosen as time. The method of carrying each into the other is an achievement of axiomatic field theory Known (§3 of the exposition by Jaffe–Witten [2000, 0-03], the official problem statement of the Clay prize problem; hereafter "the exposition").

The lattice reduces rotations to 90°

The rotations left on the lattice are only steps of 90° and reflections, and that $SO(4)$ comes back for $a\to0$ also has to be shown. For the free field on a four-dimensional lattice ($m=1$, $a=1/2n$), we compare the correlations at distance-1 points $(2n,0,0,0)$ and $(n,n,n,n)$. The continuum correlation is $G(r)=\dfrac{m\,K_1(mr)}{4\pi^2r}$. $K_1$ is a special function called the modified Bessel function, which decays as $e^{-mr}$ far away (near the origin it behaves like $1/(mr)$, $G\approx1/(4\pi^2r^2)$).

On a coarse lattice the correlation differs by a factor of about 1.8 depending on direction, and the difference vanishes in proportion to $a^2$. Wilson's lattice (5-01) keeps reflection positivity (4-10) exactly at the price of breaking rotations. The exposition writes that if a regularisation loses reflection positivity, there are few ways to recover it (Jaffe–Witten §6.5).

If $S_E$ is bounded below, $e^{-S_E}/Z$ can be read like a probability density on field configurations, and (3) are its moments. Field theory becomes a theory of random functions on four-dimensional space. The exposition attributes this view to Symanzik (Jaffe–Witten §3) Known.

Precisely, there is no Lebesgue measure $\mathcal D\varphi$ in infinite dimensions (1-09), so it is correct as it stands only in finite dimensions. On a periodic lattice $\Lambda_{a,L}$ with spacing $a$ and side $L$,

is a genuine probability measure ($S_E^{(a)}$ is (2) with derivatives replaced by differences and integrals by sums). Making a field theory means showing the limits of the moments as $a\to0$, $L\to\infty$ and their properties (6-01). The limit of the free field is a Gaussian measure on a space of distributions, and the exposition writes that nonlinear fields need non-Gaussian measures ('one requires non-Gaussian dµ to obtain nonlinear fields') (Jaffe–Witten §6.2, 1-09).

ComputedThe numbers in this figure were computed on this machine (random numbers with a fixed seed).

Looking at the samples, one cannot tell which axis is time. In 200 samples made the same way (python3), with $\xi=8$ (the mass coefficient of the action $M=2\sinh\frac1{2\xi}$, chosen so that the correlation decays exactly as $e^{-r/\xi}$), the correlation at distance 8 is 0.1120 ± 0.0032 vertically and 0.1112 ± 0.0030 horizontally, against 0.1107 exactly, and for $\xi=2,4,16$ too they agree within twice the standard error Computed. That $G(0)$ grows logarithmically with $\xi$ is a manifestation of the divergence of point values in two dimensions (2-07 §09).

$S_E[A]$ does not change under gauge transformations (3-02), so $e^{-S_E}$ is constant on gauge orbits (the collections of connections carried into one another by gauge transformations, 3-04). An orbit is infinite-dimensional, with one group attached at each point, and the integral in its direction is infinite even formally.

(A) The weight $e^{-(x-y)^2/2}$ does not change under the shift $(x,y)\to(x+c,y+c)$, and the orbits are infinitely long lines. The integral $Z_R$ over the square $[-R,R]^2$ grows in proportion to $R$, but the average of $(x-y)^2$, which is unchanged by the shift, tends to 1. (B) The orbits of the weight $e^{\beta\cos(\theta_1-\theta_2)}$ are circles of length $2\pi$, and integrating with the uniform measure on the circle (Haar measure), $Z=(2\pi)^2I_0(\beta)$ is finite for every $\beta$ (checked by double integrals and Bessel functions) Computed. Here $I_0$ is the $\nu=0$ of the $I_\nu$ of Table 2, and

Setting $\theta=\theta_1-\theta_2$, the remaining integral over $\theta_2$ just produces one factor of the orbit length $2\pi$.

In (A) the denominator and numerator diverge at the same rate, and only the ratio remains ($Z_R/2R\to\sqrt{2\pi}=2.5066$). Wilson's lattice (5-01) has the form of (B). On a finite lattice the link variables $U\in G$ are integrated with the Haar measure of a compact group, so the integral is finite without fixing the gauge ('the lattice approximation reduces the functional integration to a finite-dimensional integral', Jaffe–Witten §6.5) Known. Treating it in the continuum needs gauge fixing (4-08), and there is the obstruction that representatives cannot be chosen globally and continuously (3-04). Either way, only gauge-invariant observables (Wilson loops, $\mathrm{tr}\,F^2$ and so on) have meaning.

If there are a Hamiltonian $H$ (with vacuum energy 0) and a vacuum $\Omega$, the correlation of observables $O$ separated by imaginary time $\tau$, expanded in eigenstates $|n\rangle$, is

(an integral if the spectrum is continuous). If the vacuum is unique (the only state of energy 0 is $\Omega$), $\langle\Omega|O|\Omega\rangle=0$, and the spectrum of $H$ has nothing in $(0,\Delta)$, the correlation decays at least as $e^{-\Delta\tau}$. Example. Integrating the two-point function of the free field over space leaves only one particle with momentum 0, and the decay rate is exactly the mass Known.

This is how the gap shows on the Euclidean side; the definition is set out in 4-11. On the lattice the weight joining layers at neighbouring times is the transfer matrix $T=e^{-aH}$ (2-07 §08), and in gauge theory, in the limit where the step in the time direction goes to 0, the Kogut–Susskind Hamiltonian comes out of the gauge transfer matrix of $A_0=0$ (abstract of Creutz 1977) Known. The lattice Hamiltonian is treated in 5-03.

What this chapter can say

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Next chapter to read: 4-03 Perturbation theory and Feynman diagrams. The Euclidean propagator of (5) becomes the lines. The conditions for going back to physical spacetime are in 4-10.

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