2026-10-01 · chapter Yang–Mills and the mass gapPart 4 Quantisationreflection positivity and reconstruction

The Osterwalder–Schrader axioms and reconstruction — reflection positivity turns probability back into quantum mechanics

What can be made on the Euclidean side is a probability measure $\mu$ and its moments (4-02, One sign is different). What the problem statement asks for is a Hilbert space $\mathcal H$, $H\ge0$ and a vacuum $\Omega$ (4-09), and §4 writes 'Existence includes establishing axiomatic properties at least as strong as those cited in [45, 35].' [35] is the two papers of Osterwalder–Schrader (OS) Known.

The way there (from spacetime to imaginary time) is analytic continuation. On the way back, Schwinger said in the discussion at the 1958 CERN conference 'The question of to what extent you can go backwards, remains unanswered' (according to Jaffe 2018). Nelson built the way back under the Markov property (the property that, given the present, past and future are independent), and OS replaced it by reflection positivity (problem statement §3) Known.

Example. For one particle, $\mathbb E[x(\tau)x(0)]=\langle x\psi_0,e^{-\tau(H-E_0)}x\psi_0\rangle$ is the way there (formula (8) of 2-06). Section 06 is the reverse.

We write points as $x=(\tau,\vec x)\in\mathbb R^d$ and call $\tau$ the Euclidean time. The $n$-point Schwinger function is a moment of $\mu$; precisely, the object is $S_n(f)$ smoothed with test functions (1-11). It is related to the Wightman function formally by the substitution $x^0\to-i\tau$, $S_n=W_n(x_1^E,\dots,x_n^E)$ (Summers §3) Known.

Example: the free field of mass $m$ is the Gaussian measure with covariance $(-\Delta+m^2)^{-1}$ ($\Delta$ is the Laplacian $\sum_\mu\partial_\mu^2$, not the $\Delta$ of the mass gap) (problem statement §6.2). Integrating over $p_0$ alone (formula (11) of 2-04) gives

where $\omega=\sqrt{\vec p^{\,2}+m^2}$ is the energy of a particle of momentum $\vec p$. The energy is written into the rate at which the correlation decreases: this is the seed of reconstruction. For $d=1$ the right-hand side is itself $S_2(\tau)$, which becomes the example of section 04. The higher $S_n$ are sums of products of $S_2$ over pairings (1-08).

Let $\Theta(\tau,\vec x)=(-\tau,\vec x)$ be time reversal.

KnownTable 1 The OS conditions (scalar field; the numbering is said to be that of OS 1975)

Notation of the table. The support of a function $f$ is (the closure of) the set of points where $f\ne0$, and "supported at positive times" means $f=0$ for $\tau\le0$. $\otimes$ is the function with separated variables multiplied, $(f\otimes g)(x,y)=f(x)g(y)$, and $\Theta\overline{f}$ is $\overline{f}$ with the time of each variable reversed. $|f|_k$ is the supremum of the derivatives of $f$ up to order $k$ multiplied by polynomial weights, measuring the smoothness and decay of rapidly decreasing functions (1-11).

That (E0) excludes coinciding points agrees with the fact that $S_2$ of the free field diverges at $x=y$ (for $d\ge2$). The proof in the 1973 paper had an error concerning the continuity of tensor products of distributions (Jaffe 2018), and the second paper of 1975 is said to have given a correct proof with the linear growth condition (E0′) added as an assumption (according to secondary sources; OS 1975 bibliographic data only). It has been shown that the $S_n$ made from a Wightman theory satisfy (E0) to (E4), but they do not necessarily satisfy (E0′). So it is not known whether the OS axioms including (E0′) are equivalent to the Wightman axioms themselves (Summers §3, footnote 15). What OS 1975 showed is equivalence with a modified form of Wightman theory (Jaffe 2018; the modification is said to add a growth condition) Known. Example. The free field satisfies all of them, and (E0′) follows from $S_{2k}$ being a sum over $(2k-1)!!\le(2k)!$ pairings. The number of pairings $(2k-1)!!$ goes into $\sigma_n$, and each factor is bounded by a seminorm of fixed order.

Let $F$ be a quantity determined by the fields at positive times only (for example $F=\sum_jc_j\phi(\tau_j,\vec x_j)$, $\tau_j\gt0$), and $\Theta F$ the same with time reversed. (E2) has the following form.

Speaking in metaphor, it is a condition that measures the length of a future quantity by its overlap with its mirror image. Precisely, it is the condition that the quadratic form of correlations between past and future is not negative, and unlike $\mathbb E[|F|^2]\ge0$ it is not trivial. For a one-dimensional stationary Gaussian process (covariance $C(\tau-\tau')$), taking $F=\sum_jc_j\phi(\tau_j)$,

The right-hand side is the diagonal and determinant of the $2\times2$ matrix of two points, and requires the convexity of $\log C$. A sufficient condition: if $C(\tau)=\int_{[0,\infty)}e^{-E\tau}d\rho(E)$ ($\rho\ge0$), the left-hand side becomes $\int|\sum_jc_je^{-E\tau_j}|^2d\rho\ge0$.

Example: $e^{-\tau^2}$ is not reflection positive

$e^{-\tau^2}$ is a legitimate covariance (its Fourier transform is positive; Bochner's theorem: a continuous function $C$ is positive definite, that is, $\sum_{i,j}\overline{c_i}c_jC(\tau_i-\tau_j)\ge0$ for any $\tau_i$ and complex $c_i$, if and only if its Fourier transform is a positive measure), but $\log C$ is concave, and at $\tau_1=0.2$, $\tau_2=1.5$, from $C(0.4)=0.852144$, $C(1.7)=0.055576$, $C(3.0)=1.234\times10^{-4}$ the determinant is $-2.98\times10^{-3}$ Computed. (Why it cannot be written in this form: for positive $\rho$, $C(\tau)\ge\rho([0,E])e^{-E\tau}$, so it cannot decrease faster than an exponential. That this is the cause of the violation becomes clear once necessity is shown in section 05.)

ComputedThe numbers in this figure were computed on this machine (the matrix of (5) computed on the spot).

ComputedTable 2 "Matrix" is the smallest eigenvalue of $[C(\tau_i+\tau_j)]$ ($\tau_i=0.1i$, $i\le40$), and "two points" the minimum of the determinant of (5) for $\tau_1,\tau_2\in(0,2]$.

All six are legitimate as covariances, and the bottom four violate reflection positivity ($(a)_+=\max(a,0)$). The damped oscillation is the form of a propagator with complex poles $\omega=\pm3\pm i$, and the Gribov–Zwanziger form of 7-02 also has complex poles. By (5), a field whose correlation in the time direction becomes negative cannot be read as a field of a reflection-positive theory. In lattice calculations in the Landau gauge, the correlation of gluons in the time direction has been observed to become negative, and physics reads this as a sign that gluons are not observed particles Physics.

Given (4), the rest proceeds by linear algebra Known. (1) Inner product. On the space $\mathcal E_+$ of all bounded functions of the fields at positive times, put $\langle F,G\rangle:=\mathbb E[\overline{\Theta F}\,G]$. By (4), $\langle F,F\rangle\ge0$, but $\langle F,F\rangle=0$ is possible even with $F\ne0$ (positive semidefinite). Even so, Cauchy–Schwarz $|\langle F,G\rangle|^2\le\langle F,F\rangle\langle G,G\rangle$ holds (from the discriminant of the quadratic $\langle F+tG,F+tG\rangle\ge0$ in $t$). From this the set $\mathcal N$ of elements of length 0 is a subspace, and we complete $\mathcal E_+/\mathcal N$, which identifies elements that differ only by something of length 0, to get $\mathcal H$ (for completion, 1-10). (2) Shifting. The shift $U_s$ by $s\ge0$ in the time direction takes quantities at positive times to quantities at positive times, and from the translation invariance of $\mu$ and $\Theta U_s\Theta=U_{-s}$,

(3) Contraction. Repeating (6) and Cauchy–Schwarz $n$ times gives

and since $\|U_uF\|\le\sup|F|$, as $n\to\infty$ we get $\|U_sF\|\le\|F\|$. Length 0 goes to length 0, and $U_s$ defines $T(s)$ on $\mathcal H$. (4) Hamiltonian. $T(s)$ is symmetric with $0\le T(s)=T(s/2)^2\le1$ and $T(s)T(u)=T(s+u)$, so if it is continuous, the spectral theorem (1-10) gives

Necessity in one dimension, and real time

In one dimension $C(\tau+2\varepsilon)=\langle\psi_\varepsilon,e^{-\tau H}\psi_\varepsilon\rangle=\int e^{-E\tau}d\rho_\varepsilon(E)$ ($\psi_\varepsilon=[\phi(\varepsilon)]$, and $\rho_\varepsilon$ is the spectral measure; $\phi$ is not bounded, but for Gaussians $\mathcal E_+$ may be extended to square-integrable functions), so the sufficient condition of section 04 is also necessary. Therefore a $C$ that decreases faster than an exponential, like $e^{-\tau^2}$, cannot be reflection positive. The form $\int e^{-E\tau}d\rho(E)$ is called the Laplace transform of $\rho$, and on this side it corresponds to the characterisation of Widder 1934. The connection with reflection positivity is said to have been pointed out by Klein–Landau (Jaffe 2018). Since $H\ge0$, $e^{-zH}$ makes sense for $\mathrm{Re}\,z\ge0$, and at $z=it$ it is time evolution $e^{-itH}$. Boosts are obtained in the OS reconstruction from the analytic continuation of the Wightman functions. Fröhlich–Osterwalder–Seiler 1983 and Klein–Landau abstracted this into the form of extending the rotation angle to imaginary values (0-04 §02) (Jaffe 2018 §3.3.1). (E0′) is used at the stage of obtaining $W_n$ Known. Example. If $C=e^{-m|\tau|}/(2m)$ and $F=\phi(\tau_0)$, then $\|U_sF\|=e^{-ms}\|F\|$, and on this vector $H$ acts as multiplication by $m$.

With $H=\frac12p^2+x^4$ of Table 4 of 2-07 we make $C(\tau)=\mathbb E[x(\tau)x(0)]$ (diagonalising in a basis of 240 functions), then forget $H$ and carry out section 05 with $C$ alone. With $s=0.25$ and $F_i=x((i+\tfrac12)s)$ ($i\lt n$),

from the generalised eigenvalue problem $G^{(1)}v=\lambda Gv$ (the same as the eigenvalues of $G^{-1/2}G^{(1)}G^{-1/2}$ if $G$ is positive definite) we read $E=-\frac1s\log\lambda$. Since these are eigenvalues of $T(s)$ restricted to the subspace spanned by $F_i$, by the variational principle (1-10) they are upper bounds on the true levels Known. That $G$ is positive definite (reflection positivity) is presupposed. $x$ is odd, so what can be read are $E_1-E_0$, $E_3-E_0$, …, and the weights $|\langle\psi_0,x\psi_n\rangle|^2$ fall sharply, from 0.287 at $n=1$ to $3.9\times10^{-9}$ at $n=7$ Computed.

ComputedThe numbers in this figure were computed on this machine (results of the same calculation as Table 3).

ComputedTable 3 Levels read from the table of correlations alone (no noise), and the smallest eigenvalue of $G$.

At $n=3$ three levels come out to 4 or 5 digits, all above the true values. At $n=4$ the smallest eigenvalue of $G$ reaches the size of double-precision rounding, and beyond that nothing can be distinguished.

Detour: noisy data — the way back is unstable

For Gaussian measures the reconstruction can be written out completely. With the one-dimensional $C(\tau)=e^{-m|\tau|}/(2m)$ (an oscillator of angular frequency $m$), using Wick products, the powers of $\phi$ adjusted by the variance $\sigma^2=C(0)$ (${:}\phi^2{:}=\phi^2-\sigma^2$, ${:}\phi^3{:}=\phi^3-3\sigma^2\phi$, and in general ${:}\phi^k{:}=\sigma^k\mathit{He}_k(\phi/\sigma)$ with the Hermite polynomials $\mathit{He}_k(x)=(-1)^ke^{x^2/2}\frac{d^k}{dx^k}e^{-x^2/2}$),

and the decay rate $km$ is the level. For this process (the stationary Gaussian process with covariance $e^{-m|\tau|}/(2m)$ is called the Ornstein–Uhlenbeck process), $[{:}\phi^k{:}]$ is an eigenvector with eigenvalue $km$, and $H=m\,a^\dagger a$ Known. The levels $E_k-E_0=km$ of 2-06 (2-06) come out from the correlations alone. Example. At $m=1$, $k=2$ it is $\tfrac12e^{-2\tau}$, and the rate 2 is a two-particle state.

In $d$ dimensions, by (3) positive exponentials line up momentum by momentum, so (E2) holds (problem statement §6.2); reconstruction yields Fock space, and the spectrum is $\{0\}\cup[m,\infty)$ (2-06 §08). KnownThe action of Yang–Mills has no mass term (3-02), and the gap must be an output of the interaction (4-11).

Regularisations that suppress the ultraviolet (4-04) can break reflection positivity. Example: the Pauli–Villars type. Subtracting the propagator of a heavy mass $M$ makes it decrease fast, as $p^{-4}$, at large $p$.

It is legitimate as a covariance, but in the time direction the exponential weight is negative at $E=M$. At $m=1$, $M=3$, $CC''-C'^2=-\tfrac13e^{-4\tau}\lt0$, so $\log C$ is concave everywhere, and the test of (5) is violated at any two distinct points (the last row of Table 2) Computed. A negative weight corresponds to states with negative inner product (ghosts; a different meaning from the "ghost" of 4-08 §03, which refers to the Faddeev–Popov anticommuting fields) Known. §6.5 of the problem statement writes that there are few ways of recovering reflection positivity lost through regularisation, and gives dimensional regularisation and Pauli–Villars regularisation as examples (quoted in 0-03 §03).

The lattice is the opposite. The Wilson action keeps reflection positivity (Osterwalder–Seiler 1978, problem statement §6.5), and the transfer matrix and the Hamiltonian stand up (for an example, 0-04 §06; for gauge theories, 5-03). In exchange, rotations are cut down to steps of 90°. Reflection positivity is an inequality, so it passes to the limit (6-01 §07) Known.

Observables. What has meaning are gauge-invariant quantities, and the local ones are composite quantities such as $\mathrm{Tr}\,F_{\mu\nu}F_{\rho\sigma}(x)$ (problem statement §4). On Wilson loops (3-02), Chatterjee, citing Seiler, writes 'the key observables in these theories do not take values at points, but at curves'. Gauge fixing. Correlations of fixed fields are in general not reflection positive, and covariant gauges use spaces with indefinite inner products (4-08). Magnen–Rivasseau–Sénéor 1993 say that covariant gauges such as the Feynman gauge 'cannot be used directly at the beginning because of their lack of positivity', use a regularised axial gauge, and go no further than expecting that OS positivity 'could be shown to hold with some additional work'. Volume. Even on $T^4$, if a reflection-positive measure can be made, a finite-volume Hilbert space and transfer matrix are obtained (this is so for the periodic box of the lattice; the construction itself on the continuum $T^4$ is unfinished; problem statement §6), but a Poincaré-covariant theory on $\mathbb R^4$ needs $T^4\to\mathbb R^4$ (0-04 §09) Known.

Example. From Driver's 1989 measure in two dimensions, the Euclidean invariance of gauge-invariant expectation values follows, but the full set of axioms is not claimed (6-02). §6.4 of the problem statement calls the two-dimensional abelian Higgs model (Brydges–Fröhlich–Seiler) 'the only complete example of an interacting gauge theory satisfying the axioms' (an assessment from 2000), and §6.5 continues 'Establishing a quantum mechanical Hilbert space is part of the solution to this Millennium problem.' Known.

What this chapter can say

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Next chapter to read: 4-11 The definition of the mass gap. For reflection positivity on the lattice, 5-03.

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