Constructive quantum field theory — Lecture 1 (slides)

• Goal: construct, with full rigour, interacting relativistic quantum fields, i.e. models satisfying axioms encoding quantum mechanics, special relativity and causality.

• Status:

  • \(d = 2\): \(P (\varphi)_2\), Yukawa_\(2\), sine-Gordon, …

  • \(d = 3\): \(\Phi^4_3\), …

  • \(d = 4\): no interacting example; critical lattice \(\varphi^4_4\) has Gaussian scaling limits (Aizenman–Duminil-Copin 2021).

• Lesson of the late 1960s–70s: work in imaginary time. A QFT becomes a probability measure on \(\mathcal{S}' (\mathbb{R}^d)\), which looks like the continuum limit of a classical spin system at a critical point.

• Modern tools (this course): stochastic quantisation, singular SPDEs, variational methods; RG and cluster expansions as complements.

Conventions: \(\hbar = c = 1\), spacetime dimension \(d = 1 + s\), \(\langle f, g \rangle = \int \bar{f} g\), \(\hat{f} (k) = \int \mathrm{e}^{- \mathrm{i} k \cdot x} f (x) \mathrm{d} x\).

Relativistic quantum fields

Lecture 1

Formally \(\varphi (f) = \int \varphi (x) f (x) \mathrm{d} x\), \(\varphi (x^0, \textbf{x}) = \mathrm{e}^{\mathrm{i} Hx^0} \varphi (0, \textbf{x}) \mathrm{e}^{- \mathrm{i} Hx^0}\).

Wightman functions \(W_n (f_1, \ldots, f_n) = \langle \Omega, \varphi (f_1) \cdots \varphi (f_n) \Omega \rangle \in \mathcal{S}' (\mathbb{R}^{dn})\):

• positivity \(\sum_{j, k} W_{j + k} (g_j^{\ast} \otimes g_k) \geq 0\);Poincaré invariance;

• spectral condition: Fourier support in \(\bar{V}_{\hspace{-0.17em} +}^{\hspace{0.17em} n - 1}\) (difference variables);

• locality; cluster property \(\Leftrightarrow\) unique vacuum.

Theorem. (Wightman reconstruction; Streater–Wightman 1964) A sequence \((W_n)\) of tempered distributions with \(W_0 = 1\) satisfying positivity, invariance, spectral condition and locality is the sequence of Wightman functions of a Wightman theory, unique up to unitary equivalence.

• To construct a QFT it suffices to construct its Wightman functions.

• Difficulty: positivity involves all \(W_n\) at once and refers to real time, while what we can construct lives in imaginary time.

• The Euclidean approach replaces it by reflection positivity, checkable on lattice approximations and stable under limits.

• The axioms say nothing about dynamics: no Lagrangian; the interaction is hidden in the higher \(W_n\).

\(\omega (\textbf{p}) = (| \textbf{p} |^2 + m^2)^{1 / 2}\),\(\mathrm{d} \Omega_m (\textbf{p}) = \dfrac{\mathrm{d} \textbf{p}}{(2 \pi)^s \hspace{0.17em} 2 \omega (\textbf{p})}\) (invariant measure on \(H_m^+ = \{p^2 = m^2, p^0 > 0\}\)), Fock space over \(L^2 (\mathrm{d} \Omega_m)\):

with \([a (\textbf{p}), a^{\ast} (\textbf{q})] = (2 \pi)^s 2 \omega (\textbf{p}) \hspace{0.17em} \delta (\textbf{p} - \textbf{q})\), \(a (\textbf{p}) \Omega = 0\). Then

but

UV divergence: logarithmic for \(s = 1\), power-like for \(s \geq 2\). Not an artefact of the free field.

Proposition. (Wightman 1964) Assume (W0)–(W2) and \(\varphi (f) \Omega = \int f (x) U (x, \mathbb{1}) \psi \hspace{0.17em} \mathrm{d} x\) for some \(\psi \in \mathcal{H}\) (“\(\varphi (0) \Omega = \psi\) exists”). Then \(\psi \in \mathbb{C} \Omega\), hence \(\varphi (f) \Omega = c \hspace{0.17em} (\int f) \hspace{0.17em} \Omega\).

Lemma. For \(d \geq 2\), a finite \(\mathcal{L}^{\uparrow}_+\)-invariant measure on \(\bar{V}_{\hspace{-0.17em} +}\) is \(c \hspace{0.17em} \delta_0\).

• Combined with locality and Reeh–Schlieder: \(\varphi\) is a multiple of \(\mathbb{1}\).

• Quantum fields are operator-valued distributions: nonlinear expressions in \(\varphi (x)\) need a definition \(\Rightarrow\) renormalisation.

Lemma. A boost of rapidity \(\theta\) acts as \(p^0 \pm p^1 \mapsto \mathrm{e}^{\pm \theta} (p^0 \pm p^1)\). The sets

are mapped to each other by the boost with \(\mathrm{e}^{\theta} = 2\); they are disjoint and cover \(\bar{V}_{\hspace{-0.17em} +} \setminus \{0\}\). Invariance and finiteness force \(\mu (S^{\pm}_j) = 0\).

Proposition.

1. Covariance gives \(\int f (x) U (\Lambda x) (U (0, \Lambda) \psi - \psi) \mathrm{d} x = 0\) for all \(f\), so \(U (0, \Lambda) \psi = \psi\).

2. The spectral measure \(\mu\) of \(\psi\) for translations is finite, supported in \(\bar{V}_{\hspace{-0.17em} +}\) and Lorentz invariant, so \(\mu = \| \psi \|^2 \delta_0\).

3. Hence \(U (a) \psi = \psi\) for all \(a\), and \(\psi \in \mathbb{C} \Omega\) by uniqueness of the vacuum.

Let \(W_2^{(M)}\) be the free two-point function of mass \(M \geq 0\) and \(\langle \Omega, \varphi (f) \Omega \rangle = v \int f\).

Theorem. (Källén 1952, Lehmann 1954) Under (W0)–(W2) there is a positive, polynomially bounded measure \(\rho\) on \([0, \infty)\) (the spectral measure of the field) such that

(\(\rho (\{0\}) = 0\) if \(d = 2\).)

Sketch. \(\psi_f = \varphi (f) \Omega - v (\int f) \Omega\) gives \(W_2 - v^2\) of positive type \(\Rightarrow\) (Bochner–Schwartz) Fourier transform of a positive measure \(\mu\), supported in \(\bar{V}_{\hspace{-0.17em} +}\), \(\mu (\{0\}) = 0\), Lorentz invariant. Orbits are the hyperboloids \(H_M^+\) (and the cone for \(M = 0\)), each with a unique invariant measure \(\mathrm{d} \Omega_M\). Disintegrate along \(M\).

• Free field: \(\rho = \delta_{m^2}\). Conversely \(\rho = \delta_{m^2}\), \(m > 0\), forces \(\varphi - v\) to be free, by locality (Jost–Schroer).

• Generalised free fields \(\varphi = \sum_i c_i \varphi_{M_i}\) have \(\rho = \sum_i c_i^2 \delta_{M_i^2}\) and no interaction: \(\rho \neq \delta_{m^2}\) is necessary, not sufficient.

• Particles: an isolated atom \(Z \delta_{m^2}\) is a particle of mass \(m\); multi-particle continuum from \((2 m)^2\) (or \((3 m)^2\) with \(\varphi \mapsto - \varphi\) symmetry). Weakly coupled \(P (\varphi)_2\): Glimm–Jaffe–Spencer 1974.

• No point fields again: \(W_2^T (0) = \int \rho (\mathrm{d} M^2) \int \mathrm{d} \Omega_M = \infty\).

• Canonical fields: CCR at sharp times \(\Rightarrow\) \(\int \rho = 1\), so \(Z \leq 1\). In \(d = 4\) perturbatively \(\int \rho = \infty\) (wave-function renormalisation).

The free field solves, as an operator-valued distribution,

and its sharp-time fields \(\varphi (t, g)\), \(\pi (t, g)\), \(\pi = \partial_t \varphi\), satisfy the CCR

Interacting version: \((\Box + m^2) \varphi + \lambda \varphi^3 = 0\) together with the CCR.

\(s = 1\), space \(\mathbb{T}_L =\mathbb{R}/ L\mathbb{Z}\), momenta \(k \in \frac{2 \pi}{L} \mathbb{Z}\), \(\omega_k = (k^2 + m^2)^{1 / 2}\):

• An infinite family of independent harmonic oscillators, one per mode.

• With a UV cutoff \(|k| \leq R\):

  \(\displaystyle \langle \Omega, \varphi_R (0, x)^2 \Omega \rangle = \sum_{|k| \leq R} \frac{1}{2 L \omega_k} = \frac{1}{2 \pi} \log \frac{R}{m} + O (1) \to \infty .\)

• Wick ordering \(: \varphi_R^2 := \varphi_R^2 - \langle \Omega, \varphi_R^2 \Omega \rangle\) removes exactly this divergence (Exercise 1.7).

1. Products of distributions. \(\varphi (x)^3\) is meaningless and must be renormalised: Wick ordering suffices in \(d = 2\); more counterterms in \(d = 3\); triviality in \(d = 4\).

2. Haag's theorem. In infinite volume the interacting field at fixed time is not unitarily equivalent to the free one: there is no interaction picture.

3. Semiboundedness. Even on \(\mathbb{T}_L\), \(H = H_0 + \lambda \int : \varphi (0, x)^4 : \mathrm{d} x\) is positive plus unbounded below. Nelson 1966: bounded below (hypercontractivity). Volume cutoff removed by Glimm–Jaffe; Wightman axioms for weak coupling \(P (\varphi)_2\) by Glimm–Jaffe–Spencer 1974.

The Euclidean approach (Symanzik, Nelson, Osterwalder–Schrader, Guerra–Rosen–Simon) simplifies this enormously.

From real to imaginary time

Lecture 1

Lattice \(\Lambda_{\varepsilon} = \varepsilon \mathbb{Z}/ L\mathbb{Z}\), \(N = L / \varepsilon\) sites, \(\mathcal{H}_{\varepsilon} = L^2 (\mathbb{R}^{\Lambda_{\varepsilon}}, \mathrm{d} \varphi)\), \(\varphi_x\) multiplication, \(\pi_x = - \mathrm{i} \varepsilon^{- 1} \partial_{\varphi_x}\), so \([\varphi_x, \pi_y] = \mathrm{i} \varepsilon^{- 1} \mathbb{1}_{x = y}\).

• Heisenberg fields solve the lattice wave equation

  \(\displaystyle \partial_t^2 \varphi_x = (\Delta_{\varepsilon} \varphi)_x - m^2 \varphi_x - V' (\varphi_x)\)

  — the quantum wave equation, now rigorous.

• Schrödinger operator with confining potential: self-adjoint, compact resolvent, \(\mathrm{e}^{- tH_{\varepsilon}}\) trace class with positive kernel.

• Perron–Frobenius: simple ground state \(\Omega_{\varepsilon} > 0\), energy \(E_0\). \(\bar{H} := H_{\varepsilon} - E_0 \geq 0\).

1. Analyticity. \(\mathrm{e}^{- \mathrm{i} \zeta \bar{H}} = \int \mathrm{e}^{- \mathrm{i} \zeta \lambda} \mathrm{d} E (\lambda)\) is a contraction for \(\text{Im} \zeta \leq 0\), since \(\lambda \geq 0\): analytic in the interior, continuous up to the boundary.

2. Uniqueness. An analytic function on the half plane vanishing on \(- \mathrm{i} \mathbb{R}_+\) vanishes; iterate variable by variable.

3. Semigroup. \(\mathrm{e}^{- \tau \bar{H}}\) is a contraction semigroup, which can be represented as a path integral (Feynman–Kac).

4. Projection onto the vacuum. \(\mathrm{e}^{- T \bar{H}} \to | \Omega \rangle \langle \Omega |\) as \(T \to \infty\).

Relativistic version: Wightman functions are boundary values of functions analytic in the forward tube; Lorentz covariance extends them to all non-coincident Euclidean points (Schwinger functions are symmetric and Euclidean invariant).

Kinetic semigroup: \(\mathrm{e}^{\frac{\delta}{2 \varepsilon} \Delta} (\varphi, \varphi') = (2 \pi \delta / \varepsilon)^{- N / 2} \exp \left( - \frac{\varepsilon}{2 \delta} | \varphi - \varphi' |^2 \right)\).

Transfer matrix (symmetric Trotter step):

and \(T_{\delta}^{\lfloor \tau / \delta \rfloor} \to \mathrm{e}^{- \tau H_{\varepsilon}}\) (Trotter).

Space-time lattice \(\Lambda = (\delta \mathbb{Z}/ T\mathbb{Z}) \times (\varepsilon \mathbb{Z}/ L\mathbb{Z})\), \(T = J \delta\):

The Minkowski metric has become Euclidean.

Lattice Gibbs measure \(\mu^{T, L}_{\delta, \varepsilon} (\mathrm{d} \varphi) = Z^{- 1} \mathrm{e}^{- S_{\delta, \varepsilon} (\varphi)} \prod_u \mathrm{d} \varphi_u\):

Proposition.

Sketch. Gaussian factor \(=\) Brownian loops with diffusion \(\varepsilon^{- 1}\); \(\delta \sum_{\tau} U (\varphi_{\tau})\) is a Riemann sum for \(\int_0^T U (\omega_{\tau}) \mathrm{d} \tau\); then \(T \to \infty\) uses the spectral gap.

Vacuum expectations in imaginary time are moments of a classical probability measure.

Notes, Exercise 2.2

Rows are quantum configurations at fixed imaginary time; the transfer matrix connects consecutive rows. As \(T \to \infty\), the rows form a stationary Markov process whose one-time marginal is \(\Omega_{\varepsilon} (\varphi)^2 \mathrm{d} \varphi\).

Isotropic lattice \(\delta = \varepsilon\):

• \(\mu_{\varepsilon} \propto \prod_{\langle u, v \rangle} \mathrm{e}^{- \frac{1}{2} (\varphi_u - \varphi_v)^2} \prod_u \nu_{\varepsilon} (\mathrm{d} \varphi_u)\), \(\nu_{\varepsilon} (\mathrm{d} a) = \mathrm{e}^{- \varepsilon^2 (\frac{1}{2} m^2 a^2 + V (a))} \mathrm{d} a\).

• A ferromagnetic nearest-neighbour spin system with unbounded spins: short-range classical statistical mechanics.

• In dimension \(d\): \(\varepsilon^{d - 2} \sum_{\langle u, v \rangle}\) and \(\varepsilon^d \sum_u\).

• Nearest-neighbour \(\Rightarrow\) reflection positive: the Euclidean shadow of \(\mathrm{e}^{- \varepsilon H} \geq 0\). Conversely, RP \(\Rightarrow\) Hilbert space and \(H \geq 0\) (Exercises 2.5, 2.7).

• In lattice units the mass is \(\varepsilon m \to 0\): correlation length \(\xi \approx (\varepsilon m)^{- 1} \to \infty\) lattice spacings.

• Continuum QFT \(=\) scaling limit of short-range spin systems approaching criticality (Wilson). Conversely, RP critical models with nontrivial scaling limits give candidate QFTs.

• Griffiths–Simon: \(\varphi^4\) spins as limits of normalised sums of ferromagnetically coupled Ising spins; critical 4D Ising and \(\varphi^4_4\) have Gaussian scaling limits (ADC 2021).

Formally, as \(\varepsilon \to 0\) and \(L, T \to \infty\):

• Neither \(\mathcal{D} \varphi\) nor the action make sense: typical \(\varphi\) are distributions.

• The Gaussian part defines the Gaussian free field \(\mu_m\) (next lecture).

• Then \(\mu \propto \mathrm{e}^{- \int V_{\text{ren}} (\varphi)} \mu_m\): meaningful in finite volume for \(d = 2\) (and, with more work, \(d = 3\)).

Question. Which measures on \(\mathcal{S}' (\mathbb{R}^d)\) correspond to relativistic QFTs?

\(\hat{\mu} (f) = \int \mathrm{e}^{\mathrm{i} \varphi (f)} \mu (\mathrm{d} \varphi)\),\(\Theta (\tau, \textbf{x}) = (- \tau, \textbf{x})\).

OS0

Analyticity: \(z \mapsto \hat{\mu} (\sum_j z_j f_j)\) entire.

OS1

Regularity: \(| \hat{\mu} (f) | \leq \exp (c (\|f\|_{L^1} +\|f\|^p_{L^p}))\), \(p \in [1, 2]\), complex \(f\).

OS2

Euclidean invariance.

OS3

Reflection positivity: for real \(f_i\) supported in \(\{\tau > 0\}\), \((\hat{\mu} (f_i - \Theta f_j))_{ij} \geq 0\).

OS4

Ergodicity under time translations.

Theorem. (Osterwalder–Schrader 1973/75) OS0–OS3 \(\Rightarrow\) the moments of \(\mu\) are the Schwinger functions of a unique Wightman theory; OS4 \(\Leftrightarrow\) unique vacuum.

• \(\mathcal{E}_+ =\) functions of \(\varphi |_{\{\tau > 0\}}\), with \((F, G) =\mathbb{E}_{\mu} [\overline{\Theta F} \hspace{0.17em} G] \geq 0\) by OS3.

• Quotient by null vectors and complete: physical Hilbert space \(\mathcal{H}\).

• Time translations by \(t \geq 0\) preserve \(\mathcal{E}_+\) and descend to a self-adjoint contraction semigroup \(\mathrm{e}^{- tH}\): positivity of the energy is recovered from reflection positivity.

• Vacuum \(=\) class of \(1\); analytic continuation \(\tau \to \mathrm{i} t\) gives the Wightman functions.

Next lecture: the Gaussian free field.