# Constructive quantum field theory

A branch of mathematical physics studying the properties of models of quantum field theory. One of the problems of constructive quantum field theory consists in the investigation of interacting quantum fields in real four-dimensional space-time. Mathematically, the existence of these fields has not yet been established (1987). Therefore, the main efforts have been directed toward the study of the less singular models of quantum field theory in two- and three-dimensional space-time. Constructive quantum field theory is a synthesis of ideas and methods of axiomatic field theory and renormalization theory with modern mathematical methods. The concept of a relativistic quantum field itself admits various equivalent mathematical interpretations, enabling one to use methods from different areas of mathematics.

A quantum field can be treated either in terms of the theory of non-linear hyperbolic equations for operator-valued generalized functions, or in terms of the theory of generalized random fields (establishing a closed contact with statistical mechanics), or as a system of analytic functions of several complex variables (in the study of analytic properties of the $ S $-matrix), or it can be considered from the point of view of $ C^{\ast} $-algebras and representation theory.

In the first works on constructive quantum field theory, mainly methods from functional analysis were used. A relativistic quantum field in two-dimensional space-time satisfying the Wightman axioms was first successfully constructed ([8]) using the Euclidean formulation ([9]) of quantum field theory, enabling one to invoke methods from probability theory and statistical mechanics.

A relativistic quantum field is completely characterized by its Wightman functions $ {W_{n}}(x_{1},\ldots,x_{n}) $. The equivalent Euclidean formulation of quantum field theory is given in terms of Schwinger functions $ {S_{n}}(x_{1},\ldots,x_{n}) $ (these are obtained from $ W_{n} $ by analytic continuation to the Euclidean points $ (i x_{1}^{0},\mathbf{x}_{1},\ldots,i x_{n}^{0},\mathbf{x}_{n}) $), satisfying the **Osterwalder–Schrader axioms**. Under certain additional hypotheses, it can be proven that the $ S_{n} $’s are moments of a probability measure with special properties. The method of constructing models of quantum field theory in which one begins by constructing a probability measure and then verifies the Osterwalder–Schrader axioms for its moments has been proven to be most convenient and is the most widespread.

In the simplest case of a single scalar field, one considers a measurable space $ (\mathcal{S}'(\mathbf{R}^{d}),\mathsf{B}) $, where $ \mathcal{S}'(\mathbf{R}^{d}) $ is the space of (real-valued) tempered distributions, and $ \mathsf{B} $ is the $ \sigma $-algebra generated by the cylinder sets, and a class of probability measures $ \mu $ on $ (\mathcal{S}'(\mathbf{R}^{d}),\mathsf{B}) $ possessing the following special properties.

- On $ (\mathcal{S}'(\mathbf{R}^{d}),\mathsf{B}) $, there is defined a natural representation of the connected component $ G $ of the identity of the Euclidean group of motions of $ \mathbf{R}^{d} $ by automorphisms of the $ \sigma $-algebra $ \mathsf{B} $. It is required that the measure $ \mu $ be $ G $-invariant. This condition is the Euclidean expression of relativistic invariance.
- Let $ \Phi(f) $ denote the generalized random field on $ (\mathcal{S}'(\mathbf{R}^{d}),\mathsf{B},\mu) $ defined by $ [\Phi(f)](\omega) \stackrel{\text{df}}{=} \omega(f) $ for $ \omega \in \mathcal{S}'(\mathbf{R}^{d}) $ and $ f \in \mathcal{S}(\mathbf{R}^{d}) $. For any function $ F $ on $ \mathcal{S}'(\mathbf{R}^{d}) $, one defines $ {F_{\theta}}(\omega) \stackrel{\text{df}}{=} F(\omega_{\theta}) $, where $ {\omega_{\theta}}(f) \stackrel{\text{df}}{=} \omega(f_{\theta}) $ and $ {f_{\theta}}(x_{1},\ldots,x_{d}) \stackrel{\text{df}}{=} f(x_{1},\ldots,x_{d - 1},- x_{d}) $. Let $ \mathsf{B}_{+} $ be the $ \sigma $-algebra generated by the functions $ \Phi(f) $ with $ \operatorname{supp}(f) \subseteq \{ x \in \mathbf{R}^{d} \mid x_{d} > 0 \} $. It is required that the positivity condition of Osterwalder–Schrader holds: $$ \int {F_{\theta}}(\omega) F(\omega) ~ \mathrm{d}{\mu(\omega)} \geq 0 $$ for any $ \mathsf{B}_{+} $-measurable function $ F $ on $ \mathcal{S}'(\mathbf{R}^{d}) $. This condition expresses the positive definiteness of the scalar product in relativistic Hilbert space. For two-dimensional models, the somewhat stronger Markov condition for the field $ \Phi(f) $ is widely used.
- It is required that there exists a norm $ \| \cdot \| $ on $ \mathcal{S}(\mathbf{R}^{d}) $ such that $$ \int e^{\Phi(f)} ~ \mathrm{d}{\mu} $$ is uniformly bounded and continuous with respect to $ \| \cdot \| $ on $$ \{ f \in \mathcal{S}(\mathbf{R}^{d}) \mid \| f \| \leq 1 \}. $$
- The subgroup of translations of the group $ G $ must act ergodically. This expresses the uniqueness of vacuum in relativistic theory.

If the measure $ \mu $ satisfies the Conditions (1)–(4), then it is called a **quantum measure**, and the corresponding generalized random field $ \Phi(f) $ is called a **Euclidean field**. The moments $ S_{n} $ of $ \mu $, defined by
$$
\forall f_{1},\ldots,f_{n} \in \mathcal{S}(\mathbf{R}^{d}): \qquad
{S_{n}}(f_{1},\ldots,f_{n}) \stackrel{\text{df}}{=} \int \Phi(f_{1}) \cdots \Phi(f_{n}) ~ \mathrm{d}{\mu},
$$
are called the **Schwinger functions**, and they satisfy the Osterwalder–Schrader axioms. There exists a unique relativistic quantum field satisfying all the Wightman axioms and such that the analytic continuations of its Wightman functions to the Euclidean points are the same as the Schwinger functions of $ \mu $. If $ \mu $ satisfies Conditions (1)–(3) only, then Conditions (1)–(4) hold for all of its ergodic components.

One class of quantum measures is easily constructed, namely, the **Gaussian measures** (which depend on a parameter $ m > 0 $) with characteristic functional
$$
\int e^{i \Phi(f)} ~ \mathrm{d}{\nu_{m}} = e^{\frac{1}{2} \langle f,(- \Delta + m)^{- 1} f \rangle_{{L^{2}}(\mathbf{R}^{d})}},
$$
where $ \Delta $ denotes the Laplace operator on $ \mathbf{R}^{d} $. The corresponding Euclidean field is called the **free (scalar) Euclidean field of mass $ m $**.

The construction of non-Gaussian measures presents great difficulties, and the results depend essentially on the dimension $ d $. The usual procedure is as follows. A function $ V(\Lambda,\kappa) $ is constructed on $ \mathcal{S}'(\mathbf{R}^{d}) $ (the interaction potential) that depends on the parameters $ \Lambda $, called the **volume cut-off**, and $ \kappa $, called the **ultraviolet cut-off**. Heuristically, $ V = - \mathsf{L}_{\text{int}} $ (see the article on quantum field theory). Also, $ V(\Lambda,\kappa) $ is additive in $ \Lambda $ but not in $ \kappa $. The measure $ \mu_{\Lambda,\kappa} $ defined by
$$
\mathrm{d}{\mu_{\Lambda,\kappa}} \stackrel{\text{df}}{=}
\frac{e^{- V(\Lambda,\kappa)} ~ \mathrm{d}{\nu^{\Lambda}_{m}}}{\displaystyle \int e^{- V(\Lambda,\kappa)} ~ \mathrm{d}{\nu^{\Lambda}_{m}}}
$$
is then constructed (the definition of $ \nu^{\Lambda}_{m} $’s is given below), and one studies the limit of the sequence $ (\mu_{\Lambda,\kappa}) $ of measures as $ \Lambda,\kappa \to \infty $. For certain potentials $ V $, the limit measure $ \mu $ satisfies Conditions (1)–(4). Convergence of the measures is usually understood in the sense of the convergence of all of their moments and characteristic functionals.

For example, for models with interaction $ \lambda \Phi^{4} $ with $ d = 2 $, this procedure is made concrete in the following way. Let $ \nu^{\Lambda}_{m} $ denote a Gaussian measure on $ (\mathcal{S}'(\mathbf{R}^{2}),\mathsf{B}) $ with the characteristic functional $$ \int e^{i \Phi(f)} ~ \mathrm{d}{\nu^{\Lambda}_{m}} = e^{- \frac{1}{2} \langle f,(- \Delta_{\Lambda} + m)^{- 1} f \rangle_{{L^{2}}(\mathbf{R}^{2})}}, $$ where $ \Delta_{\Lambda} $ denotes a self-adjoint extension of the Laplace operator on $ \mathbf{R}^{2} $ with certain boundary conditions on the boundary $ \partial \Lambda $ of the domain $ \Lambda \subseteq \mathbf{R}^{2} $ (usually, $ \Lambda $ is a rectangle); the kernel $ {(- \Delta + m^{2})^{- 1}}(x,y) $ can, for example, be the Green’s function for the Dirichlet problem. Suppose further that $ h_{\kappa} \in \mathcal{S}(\mathbf{R}^{2}) $ and that $ {h_{\kappa}}(x - y) \to \delta(x,y) $ as $ \kappa \to \infty $. Then the random variable $ {\Phi_{\kappa}}(x) \stackrel{\text{df}}{=} \Phi({h_{\kappa}}(x - \bullet)) $ is, for $ \kappa < \infty $, a smooth function of the parameter $ x \in \mathbf{R}^{2} $, and $$ \lim_{\kappa \to \infty} \int {\Phi_{\kappa}}(x) f(x) ~ \mathrm{d}{x} = \Phi(f). $$ Set $$ V(\Lambda,\kappa) \stackrel{\text{df}}{=} \lambda \int_{\Lambda} \mathopen{:} {\Phi_{\kappa}^{4}}(x) \mathclose{:} ~ \mathrm{d}{x}, \qquad \lambda \geq 0, $$ where $ \mathopen{:} {\Phi_{\kappa}^{4}}(x) \mathclose{:} $ is a Wick power of $ {\Phi_{\kappa}}(x) $ (see the article on Fock spaces). Then $ e^{- V(\Lambda,\kappa)} \in {L^{p}}(\mathcal{S}'(\mathbf{R}^{2}),\mathsf{B},\nu^{\Lambda}_{m}) $, where $ p \geq 1 $. Setting $$ Z_{\kappa,\Lambda} \stackrel{\text{df}}{=} \int e^{- V(\Lambda,\kappa)} ~ \mathrm{d}{\nu^{\Lambda}_{m}} \neq 0, $$ we finally let $$ \mathrm{d}{\mu_{\Lambda,\kappa}} \stackrel{\text{df}}{=} \frac{e^{- V(\Lambda,\kappa)} ~ \mathrm{d}{\nu^{\Lambda}_{m}}}{Z_{\kappa,\Lambda}}. $$

As $ \kappa \to \infty $ and $ \Lambda \to \mathbf{R}^{2} $, all moments and characteristic functionals of the measures $ \mu_{\Lambda,\kappa} $ converge to the moments and characteristic functionals of a certain quantum measure $ \mu $. It turns out that for sufficiently large $ \lambda $, the measures $ \displaystyle \mu_{\Lambda} = \lim_{\kappa \to \infty} \mu_{\Lambda,\kappa} $ with different boundary conditions on $ \partial \Lambda $ have, in general, different limits as $ \Lambda \to \mathbf{R}^{2} $. In this case, it is said that there is a **phase transition**.

The central problem of constructive quantum field theory consists of describing all the quantum measures (phases) corresponding to a given interaction potential, and in studying the properties of the corresponding relativistic quantum fields. In the first instance, one is interested in the spectral properties of the mass operator of the Poincaré group (the study of which reduces to examining the behavior of the Schwinger functions at large distances) and the properties of the $ S $-matrix such as its analyticity, unitarity, etc. (the $ S $-matrix is studied by analytic continuation of the Schwinger functions).

The existence of quantum measures for $ d = 2 $ was proved (1978) for interaction potentials $ V = P(\Phi) $, where $ P $ is any polynomial that is bounded from below, $ V = \lambda \cos(g \Phi) $ (the sine-Gordon equation) and certain other non-polynomial interactions, as well as for some multi-components fields $ \Phi = (\Phi_{1},\ldots,\Phi_{N}) $. For a sufficiently weak polynomial interaction, a study has been undertaken of the dependence of the spectrum of the mass operator on the form of the polynomial, and the existence of the $ S $-matrix has been established. Fermion and scalar fields with Yukawa interaction have also been investigated. The Euclidean fermion field is not a generalized random field and takes values in a Grassmann algebra. However, one can "integrate out" the fermion variables, and the problem then reduces to estimating certain path integrals with respect to ordinary Gaussian measures. All these models have phase transitions for certain values of the parameters.

The constructions of relativistic quantum fields described above lead only to the so-called **vacuum sectors**, that is, to quantum fields satisfying the Wightman axioms, supplemented by the axiom of existence of vacuum. These fields are solutions of non-linear equations with obvious initial conditions. For a number of two-dimensional models (sine-Gordon, etc.), soliton sectors have been constructed in which vacuum is absent but which have a discrete spectrum for the mass operator; from the physical point of view, this is of great interest. These new sectors are constructed by means of special automorphisms of the $ C^{\ast} $-algebra of observables in the vacuum sector.

For $ d = 3 $, existence has been proven of quantum measures in a model with $ \lambda \langle \Phi,\Phi \rangle^{2} $ interaction, where $$ \langle \Phi(x),\Phi(x) \rangle = \sum_{i = 1}^{N} {\Phi_{i}}(x) {\Phi_{i}}(x). $$ Here, the interaction potential has the form $$ V(\Lambda,\kappa) = \lambda \int_{\Lambda} \mathopen{:} \langle {\Phi_{\kappa}}(x),{\Phi_{\kappa}}(x) \rangle^{2} \mathclose{:} ~ \mathrm{d}{x} + \lambda^{2} \delta m_{\kappa} \int_{\Lambda} \mathopen{:} \langle {\Phi_{\kappa}}(x),{\Phi_{\kappa}}(x) \rangle \mathclose{:} ~ \mathrm{d}{x} + C_{\kappa,\Lambda}, $$ where $ \delta m_{\kappa} $ and $ C_{\kappa,\Lambda} $ are certain definite functions of $ \kappa $ and $ \Lambda $; that is, counter terms are added. Also, in this model, a phase transition occurs for sufficiently large $ \lambda $, and it is accompanied by an $ O(N) $-symmetry breakdown.

#### References

[1] | N.N. Bogolyubov, A.A. Logunov, I.T. Todorov, “Introduction to axiomatic quantum field theory”, Benjamin (1975). (Translated from Russian) |

[2] |
Constructive quantum field theory, Lect. notes in physics, 25, Springer (1973). |

[3] |
M. Reed, B. Simon, “Methods of modern mathematical physics”, 1–4, Acad. Press (1972–1978). |

[4] | B. Simon, “The $ P(\varphi)_{2} $ model of Euclidean (quantum) field theory”, Princeton Univ. Press (1974). |

[5] |
C. Hepp, “Théorie de la renormalisation”, Lect. notes in physics, 2, Springer (1969). |

[6] |
Euclidean quantum field theory. The Markov approach, Moscow (1978). (In Russian; translated from English) |

[7] |
R.L. Dobrushin, R.A. Minlos, “Polynomials of linear random functions”, Uspekhi Mat. Nauk, 32: 2 (1977), pp. 67–122. (In Russian) |

[8] |
J. Glimm, A. Jaffe, T. Spencer, “The Wightman axioms and particle structure in the $ P(\varphi)_{2} $ quantum field model”, Ann. of Math., 100: 3 (1974), pp. 585–632. |

[9] |
E. Nelson, “Construction of quantum fields from Markoff fields”, J. Funct. Anal., 12: 1 (1973), pp. 97–112. |

[10] |
J. Fröhlich, “Schwinger functions and their generating functionals II. Markovian and generalized path space measures on $ \mathcal{S}' $”, Adv. Math., 23: 2 (1977), pp. 119–181. |

#### References

[a1] | J. Glimm, A. Jaffe, “Quantum physics, a functional integral point of view”, Springer (1981). |

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Constructive quantum field theory.

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