Keywords

1 Introduction

It is a great pleasure to present a contribution to this volume dedicated to Ludwig Streit, on the occasion of his 75th birthday. I have written before in [3] about Ludwig’s work and the strong influence it had on my own development. In the present paper I will concentrate on some aspects relating quantum theory with stochastic analysis and infinite dimensional analysis, stressing interactions between these areas and mentioning some open problems.

In Sect. 2 I shall briefly describe the relations between oscillatory and probabilistic integrals.

In Sect. 3 I shall describe canonical quantum mechanics from this point of view.

In Sect. 4 I shall discuss the complex relations between canonical quantum fields and S(P)DEs, and also relate them with Euclidean (and relativistic) quantum fields.

In Sect. 5 I shall provide some remarks on related systems.

2 Oscillatory and Probabilistic Integrals

Since the work of Feynman (preceded in part by Wentzell and Dirac), see, e.g., [2, 107–110], a very fruitful approach to quantum mechanics and quantum field theory is based on the consideration of heuristic complex-valued measures of the form

$$\displaystyle{ \mu _{F}(\mathrm{d}\gamma ) = \text{\textquotedblleft }Z^{-1}e^{iS(\gamma )}\,\mathrm{d}\gamma \text{\textquotedblright } }$$
(1)

on a “state space” Γ, hence γ ∈ Γ, with an “action functional” \(S(\gamma ): \Gamma \rightarrow \mathbb{R}\) associated with an underlying “classical system”, dγ a “flat measure on Γ”, i the imaginary unit and Z a normalization constant.

Physical quantities of interest are then computed as “averages” (linear functionals, integrals)

$$\displaystyle{ \int _{\Gamma }f(\gamma )\mu _{F}(\mathrm{d}\gamma ) }$$
(2)

of suitable “observable functionals” \(f: \Gamma \rightarrow \mathbb{C}\).

For example, to solve the Schrödinger equation for a non relativistic (scalar) quantum mechanical particle moving for a time from the interval [0, t], t > 0 on a D-dimensional manifold M one takes as Γ a space of paths from [0, t] into M.

For a quantum (scalar) field, vibrating for a time from the interval [0, t] and in a spatial (d − 1)-dimensional domain \(\Lambda \subset \mathbb{R}^{d-1}\) (d being then the space-time-dimension) one takes Γ as a space of paths from [0, t] into a space of real-valued (generalized) functions depending on a space variable x ∈ Λ.

Analogous choices are made, e.g., for quantum gauge fields (with Γ a space of connection 1-forms in a principal fibre bundle over a Lorentzian manifold) and for quantum gravity (with Γ a space of locally Lorentzian space-time metrics).

In the case of scalar quantum fields, f could be a product of \(n \in \mathbb{N}\) field operators \(\prod _{i=1}^{n}\gamma (t_{i},x_{i})\), \(t_{i} \in \mathbb{R}\), \(x_{i} \in \mathbb{R}^{d-1}\), in which case above integrals would yield “correlation functions” (Wightman functions) associated with the quantum field \(\gamma (t,x) \in \mathbb{R}\), \(t \in \mathbb{R}\), \(x \in \mathbb{R}^{d-1}\). Similar choices of relevant observable functions can be given in the other cases alluded above. See, e.g., [1, 2, 9, 33, 48, 52, 78, 80, 107, 109, 110], and references therein.

The choice of the action functional S depends on the problem at hand. In above examples, S has a typical Lagrangian form, e.g., in the case of a non relativistic particle (of unit mass) moving during the time interval [0, t] in the D-dimensional Euclidean space \(\mathbb{R}^{D}\) under the force given by a (continuous) potential (function) \(v: \mathbb{R}^{D} \rightarrow \mathbb{R}\) one has:

$$\displaystyle\begin{array}{rcl} S(\gamma )& =& S_{QM}(\gamma ) = \frac{1} {2}\int _{0}^{t}\left \vert \dot{\gamma }(s)\right \vert ^{2}\,\mathrm{d}s -\int _{ 0}^{t}v(\gamma (s))\,\mathrm{d}s\text{,}{}\end{array}$$
(3)

and for scalar quantum fields moving at arbitrary times \(s \in \mathbb{R}\) over the space \(\mathbb{R}^{d-1}\) one has

$$\displaystyle\begin{array}{rcl} S(\gamma ) = S_{QF}(\gamma )& =& \frac{1} {2}\int _{\mathbb{R}^{d}}\dot{\gamma }(s,\vec{x})^{2}\,\mathrm{d}s\,\mathrm{d}\vec{x} -\frac{1} {2}\int _{\mathbb{R}^{d}}\left \vert \nabla \gamma (s,\vec{x})\right \vert ^{2}\,\mathrm{d}s\,\mathrm{d}\vec{x}{}\end{array}$$
(4)
$$\displaystyle\begin{array}{rcl} & -& \frac{m^{2}} {2} \int _{\mathbb{R}^{d}}\left \vert \gamma (s,\vec{x})\right \vert ^{2}\,\mathrm{d}s\,\mathrm{d}\vec{x} -\int _{ \mathbb{R}^{d}}v(\gamma (s,\vec{x}))\,\mathrm{d}s\,\mathrm{d}\vec{x}\text{,} \\ \end{array}$$

\((s,\vec{x}) \in \mathbb{R} \times \mathbb{R}^{d-1}\), m ≥ 0.

From the quantum description given in terms of (1), (2), (3), and (4) one obtains heuristically the classical (particle resp. field) mechanical one by inserting the (reduced) Planck constant \(\hslash \) at the right place, i.e. replacing S by \(\frac{1} {\hslash } S\) and making corresponding changes in the observable fn in (2), and considering \(\hslash \) to be small.

Heuristically then from (2) with such replacements one should be able to derive from Feynman’s heuristic quantum description (1) and (2) the corresponding classical description in the limit \(\hslash \downarrow 0\), with its corresponding “small \(\hslash \)-corrections”. In the case of relativistic models, the velocity of light cn appears as a parameter. Then one should be able to recover the non relativistic limit of the relativistic quantum description (1), (2), (3), and (4) by letting the velocity of light c go to + ∞n (c occurs implicitly in S and fn). In principle one could proceed in similar ways to describe the limiting dependence of the quantum averages (2) on other parameters occurring in the action functionals and observables occurring in the examples mentioned above.

There are well known difficulties in trying to transform this heuristic approach into a rigorous one. E.g., even in the case v ≡ 0n (the “free field” case), in the computation of (2), the covariance of the “complex Gaussian measure” given by the covariance S Q F (γ) |  v = 0 appears. It is, for m 2 = 1n, the fundamental solution \((\square - 1)^{-1}(s,\vec{x};s';.\vec{x}')\), which has a singularity on the submanifold \((s - s')^{2} - (\vec{x} -\vec{ x}')^{2} = 1\), indicating that it should be interpreted as a generalized function, which then means that the relevant γ’s in (2) also are generalized functions. This creates difficulties in giving mathematical meaning to the complex-valued non linear function f(γ)n appearing in (2). In the linear case where v ≡ 0n, taking e.g. f to be of the Wick product type, a sense can easily be given to (2). However, for v ≢ 0n and neither linear nor quadratic, v(γ)n is ill defined for γ which are typical relative to the complex Gaussian measure μ F n for the case v ≡ 0.

There is a well known trick (going back to Nagumo, Schwinger, and Symanzik) of analytically continuing μ F n to a corresponding “Euclidean” probability measure (or functional) μ E heuristically written as

$$\displaystyle{ \mu _{E}(\mathrm{d}\gamma ) = \text{\textquotedblleft }Z^{-1}e^{-S_{E}(\gamma )}\mathrm{d}\gamma \text{\textquotedblright } }$$
(5)

with S E (γ)n the “Euclidean” action (in contrast to the above relativistic or “Minkowskian” action (4))

$$\displaystyle\begin{array}{rcl} S_{E}(\gamma )& =& \frac{1} {2}\iint \left [\left \vert \nabla _{s,\vec{x}}\gamma (s,\vec{x})\right \vert ^{2} + \frac{m^{2}} {2} \iint \left \vert \gamma (s,\vec{x})\right \vert ^{2}\right ]\,\mathrm{d}s\,\mathrm{d}\vec{x} +\iint v(\gamma (s,\vec{x}))\,\mathrm{d}s\,\mathrm{d}\vec{x}\text{,}{}\end{array}$$
(6)

in which case the typical γn’s in (5) become still heuristically generalized random fields but at least μ E n would look as a “positive measure”. This however, only really solves the problem for v ≡ 0 and arbitrary dn, or special v ≢ 0 when d = 2, 3. In this case one interprets μ E n as a probabilistic (σ-additive) Gaussian measure, e.g., with support on the Schwartz tempered distributions space \(S'(\mathbb{R}^{d})\). For the mentioned special cases of v ≢ 0n and d = 2, 3n, one manages, in fact, to “renormalize” the nonsensical \(v(\gamma(s,\vec{x}))\) to give it a sense as a genuine generalized random field, obtaining non trivial quantum averages (2) (heuristically \(\mu _{F}\not =\mu _{F}\vert _{v=0}\), that is, in particular, states which are typically with respect to one of the measures are atypical with respect to the other measure). Moreover from such averages one recovers by “analytic continuation” in the t variable a solution of the original problem (1) and (2).

This was achieved in connection with the programme of constructive quantum field theory [9, 71, 76, 78, 98, 106], to which Ludwig Streit contributed substantially and in various ways, see, e.g., [16–18, 65, 78, 79, 81, 101, 102, 107]. Independently of the proper methods of constructive quantum field theory, the original programme of giving directly a meaning to (1) and (2) has also been solved in some cases. E.g., μ F and μ E can be defined for various classes of (regularized) v in low space-time dimension d. The integral ∫ f(γ) dμ F (γ) is then realized as

$$\displaystyle{ _{(S)}\langle \langle f,\Phi _{\mu _{F}}\rangle \rangle _{(S)'} }$$
(7)

in the framework of the infinite dimensional distributional setting

$$\displaystyle{ (S) \subset L^{2}(\mu _{ G}) \subset (S)'\text{,} }$$
(8)

with pairing 〈〈 , 〉〉 and (S), (S)′ infinite dimensional analogues of the test functions space \(S(\mathbb{R}^{d})\) resp. tempered distributions space \(S'(\mathbb{R}^{d})\). μ G is the Gaussian white noise measure on \(\mathcal{S}'(\mathbb{R}^{d})\), defined by its Fourier transform

$$\displaystyle{ \hat{\mu }_{G}(g):=\int _{\mathcal{S}'(\mathbb{R}^{d})}e^{i\langle g,\gamma \rangle }\,\mathrm{d}\mu _{ G}(\gamma ) = e^{-\frac{1} {2} \|g\|_{L^{2}(\mathbb{R}^{d})}^{2} }\text{,} }$$
(9)

with 〈g, γ〉 the distributional pairing between \(g \in \mathcal{S}(\mathbb{R}^{d})\) and \(\gamma \in \mathcal{S}'(\mathbb{R}^{d})\).

Heuristically thus

$$\displaystyle{ \mu _{G}(\mathrm{d}\gamma ) = \text{\textquotedblleft }Z^{-1}e^{-\frac{1} {2} \int _{\mathbb{R}^{d}}\left \vert \gamma (x)\right \vert ^{2}\,\mathrm{d}x }\,\mathrm{d}\gamma \text{\textquotedblright }\text{,} }$$
(10)

and, e.g.,

$$\displaystyle{ \Phi _{\mu _{F}}(\gamma ) = e^{iS(\gamma )}e^{\frac{1} {2} \int _{\mathbb{R}^{d}}\left \vert \gamma (x)\right \vert ^{2}\,\mathrm{d}x }\mu _{G}(\mathrm{d}\gamma ). }$$
(11)

For d = 1, i.e. for quantum mechanics (anharmonic oscillator for m > 0), the functional \(\Phi _{\mu _{F}}\) in (7) and (11) is well defined for a large class of continuous v and works for solving e.g. the problem of the Schrödinger equation with potential v and with smooth resp. δ-initial conditions (Green’s function) by work by L. Streit and coworkers, see e.g. [74, 75] and, for a related approach [94].

Another approach to the rigorous construction of (1) and (2) (for d = 1, and also for d ≥ 2, but the latter with regularized v(γ)-term in (4) resp. (6)) is presented in [26].

In the latter reference also work on the detailed expansion of (2) in fractional powers of \(\hslash \) around \(\hslash = 0\) by a rigorous method of stationary phase is described (for the case d = 1).

The relations between the white noise and other approaches are discussed, e.g., in [26, 39, 93, 94].

The analogues of Wightman functions in the Euclidean setting, called Schwinger functions (they are the moment functions to μ E ), for d = 1, i.e. related to quantum mechanics, have also been described in terms of white noise functionals, see [74, 75, 94]. Corresponding results for the case v ≡ 0 and all d, or for some v ≢ 0 and d = 2, 3 will be discussed in the next Sects. 3 and 4.

For characterization of interesting probability measures as white noise functionals (like μ E and related measures) see, e.g., [88, 101, 102].

3 Canonical Quantum Mechanics

In the canonical approach to quantum mechanics (and resp. the theory of quantum fields) one takes the “Schrödinger representation” of the Hilbert space of states and operators acting in \(L^{2}(\mathbb{R}^{D},\mu )\) resp. \(L^{2}(S'(\mathbb{R}^{d}),\mu )\), for some probability measures μ.

By work of Heisenberg, Coester, Haag, Araki from the early 1960s one uses, e.g., for the D-dimensional harmonic oscillators, that \(L^{2}(\mathbb{R}^{D})\) is isomorphic to \(L^{2}(\mathbb{R}^{D},\mu )\), where \(\mu (\mathrm{d}x) = N(0; \nVdash )(\mathrm{d}x) = (\sqrt{2\pi })^{-D}e^{-\frac{1} {2} \left \vert x\right \vert ^{2} }\,\mathrm{d}x\) is the canonical Gaussian measure on \(\mathbb{R}^{D}\). E.g., for D = 1 the Hamiltonian H given on smooth functions by \(H = -\frac{1} {2}\Delta + \frac{1} {2}x^{2} -\frac{1} {2}\) acting in \(L^{2}(\mathbb{R})\) is then unitarily equivalent to the operator H μ  = −L μ given on smooth functions by \(H_{\mu } = -\frac{1} {2}\Delta + x \cdot \nabla \) in \(L^{2}(\mathbb{R},\mu )\).

L μ can be interpreted probabilistically as the restriction to smooth functions of the Ornstein-Uhlenbeck operator associated with the classical Dirichlet form \(\frac{1} {2}\int \nabla u \cdot \nabla v\,\mathrm{d}\mu\), for u, v in a suitable dense subset of \(L^{2}(\mathbb{R},\mu )\). For classical Dirichlet forms (on \(\mathbb{R}^{D}\) and also on infinite dimensional spaces) see, e.g., [4, 7, 8, 10, 28, 34, 37, 49, 57, 59, 66, 69, 72, 92]. Through the replacement of \(N(0; \nVdash )\) by a more general, not necessarily Gaussian probability measure μ, one can set up a corresponding transformation for the generator L μ (called Dirichlet operator) associated with the classical Dirichlet form \(\frac{1} {2}\int _{\mathbb{R}^{D}}\nabla u \cdot \nabla v\,\mathrm{d}\mu\) in \(L^{2}(\mathrm{R}^{D},\mu )\) to a generator − H, H ≥ 0 in \(L^{2}(\mathbb{R}^{D})\).

This “ground state transformation” was first studied in details in [27], where a general and powerful approach to singular interactions in quantum mechanics on \(\mathbb{R}^{D}\) via Dirichlet forms has been initiated. E.g., 3 nucleons interacting via 2-particle δ-functions in \(\mathbb{R}^{3}\) can be rigorously defined and discussed in this approach. This Dirichlet form approach yields, more generally, a unified picture of quantum mechanics suited both for singular interactions and infinite dimensional extensions (the analogues of above classical Dirichlet forms being naturally also defined on infinite dimensional spaces; the latter forms describe namely quantum fields, see Sect. 4).

Remark 1

The probabilistic technique of subordination permits to treat quantum systems with relativistic kinematics in a way similar to those with nonrelativistic kinematics, see [6, 44–46, 50].

To general classical Dirichlet forms, there are (properly) associated good Markov processes X = (X t ), t ≥ 0, see [69]. Intuitively these can be seen as Brownian motions distorted by a drift term given by the logarithmic derivative of μ, μ itself constitutes then an invariant measure for X t . In particular X t can be extended to a μ-symmetric process for all \(t \in \mathbb{R}\).

In the quantum mechanical situation corresponding to (4) with d = 1, n = 1, and D = 1, the path space measure μ E for X t , \(t \in \mathbb{R}\) (giving the joint distribution of \((X_{t_{1}},\ldots,X_{t_{n}})\), \(t_{i} \in \mathbb{R}\), i = 1, …, n) is heuristically given by μ E as specified in (5) and (6) with a suitable v. If we write \(S^{0}(\gamma ):= \frac{1} {2}\int \left [\left \vert \dot{\gamma }\right \vert ^{2}(s) + \left \vert \gamma (s)\right \vert ^{2}\right ]\,\mathrm{d}s\), then

$$\displaystyle{ \mu _{E}(\mathrm{d}\gamma ) = Z^{-1}e^{-S^{0}(\gamma ) }e^{-\int v(\gamma (s))\,\mathrm{d}s}\,\mathrm{d}\gamma. }$$
(12)

This can be rigorously defined for suitable v, e.g. as a probability measure on \(\mathcal{S}'(\mathbb{R}^{d})\). An analytic continuation in time of the (time ordered) moment functions given by μ E (“Schwinger functions”) permits to go over to the corresponding quantities in terms of μ F (“Wightman functions”), e.g. for v resp. a polynomial or exponential function. This yields the full information on the dynamical quantum mechanical “P(φ)1” resp. “\(\exp (\varphi )_{1}\)”-“models”. Such models can also be expressed conveniently in terms of the white noise functionals mentioned in Section 2 ([74]).

One proves that both quantum mechanical Schwinger and Wightman functions can be reformulated as limits of quantities defined in the canonical Hilbert space \(L^{2}(\mathbb{R},N(0,1))\), since for v = 0 the restriction of μ E to the σ-algebra generated by X 0 is just μ = N(0, 1) (X t acts just in the same L 2-space \(L^{2}(\mathbb{R},N(0,1))\) where also H μ and the CCR are realized) and the inclusion of the v-term can be handled by “perturbation and passage to the limit”, see, e.g., [22, 23, 27, 55, 111].

Let us summarize the situation in quantum mechanics for a particle moving in \(\mathbb{R}^{D}\). As a zero step (“Level 0”) we have a (fixed time) Hilbert space, say \(L^{2}(\mathbb{R}^{D})\), an Hamiltonian H, as well as position and momentum operators, with the relative canonical commutation relations (CCR), and a fixed time distribution μ on \(\mathbb{R}^{D}\) (that exists if the lower end of the spectrum of H is an isolated simple eigenvalue) which yields the natural Hilbert space \(L^{2}(\mathbb{R}^{D},\mu )\). At a first step (“Level 1”) one has a Markov process X t (diffusion) with values in \(\mathbb{R}^{D}\), t being the time parameter, its path space measure μ E (on, say, \(\mathcal{S}'(\mathbb{R}^{D})\)), and its moments functions (Schwinger functions). This process has μ as invariant measure. In the case v ≡ 0, m = 1, the path measure is \(\mu _{E} = N\left (0;C^{-1}\right )\), with C the operator \(\left (-\frac{d^{2}} {dt^{2}} + 1\right )1\!\!1_{D}\) (on \(\mathcal{S}'(\mathbb{R}, \mathbb{R}^{D})\)), X t is the \(\mathbb{R}^{D}\)-valued Ornstein-Uhlenbeck process satisfying \(\mathrm{d}X_{t} = -X_{t}\,\mathrm{d}t +\mathrm{ d}W_{t}\), with W t a D-dimensional Brownian motion. It is associated with the classical Dirichlet form \(\frac{1} {2}\int _{\mathbb{R}^{D}}\nabla u \cdot \nabla v\,\mathrm{d}\mu\). For D = 1 this coincides with the measure μ E given by (5) and (6), for d = 1 and m = 1. For v ≡ 0 one has \(\mu _{E} = N(0;(-\Delta + 1)^{-1})\) with Δ the Laplacian on \(\mathbb{R}\).

At “Level 2” one has another Markov process (diffusion) Y τ , τ ≥ 0. For D = 1 this has μ E (as just described at level 1 given by the D = 1, m = 1 version of (5) and (6)), as invariant measure. The Markov process is given by the stationary solution of the SDE \(\mathrm{d}Y _{\tau }(t) = \left ( \frac{\mathrm{d}^{2}} {\mathrm{d}t^{2}} - 1\right )Y _{\tau }(t)\,\mathrm{d}\tau - v'(Y _{\tau }(t))\,\mathrm{d}\tau +\mathrm{ d}W_{\tau }(t)\), τ ≥ 0, with dW τ (t) a (τ, t)-Gaussian white noise, \(\tau \in \mathbb{R}_{+}\), \(t \in \mathbb{R}\).

τ can be interpreted as a “computer time” (t is the “time” in the Euclidean version of quantum mechanics; it will be replaced by a space-time variable in \(\mathbb{R}^{d}\), d ≥ 2, in a quantum field interpretation, see Sect. 4, the quantum (particle) mechanics corresponding thus to the case d = 1. The equation for Y τ is called “stochastic quantization equation”. It is a particular case of Parisi-Wu’s approach [100] for computing invariant measures of the type of μ E .

Note that Y τ can be looked upon (as a modification of) the standard diffusion process associated with the classical Dirichlet form \(\frac{1} {2}\int _{S'(\mathbb{R}^{D})}\nabla u\nabla v\,\mathrm{d}\mu _{E} = \frac{1} {2}\ _{(S)}\langle \langle \nabla u\nabla v,\mu _{E}\rangle \rangle _{(S)'}\) given by μ E .

The determination of the associated generator on smooth cylinder functions has been given in [31, 32, 87] (based on previous results in [84, 85]).

We shall now briefly go over to describe similar Level 0–2 structures for quantum fields, i.e. associated with (5) and (6) (with non trivial dependence on the space variable \(\vec{x} \in \mathbb{R}^{d-1}\), d ≥ 2).

4 Canonical Quantum Fields and S(P)DEs

We shall only discuss, for simplicity, scalar fields.

At Level 0 we have as the analogue of the position operators resp. momentum operators in quantum mechanics, the time zero quantum field \(\varphi (\vec{x})\), \(\vec{x} \in \mathbb{R}^{d-1}\) resp. the time zero momentum field \(\Pi (\vec{x})\), satisfying the canonical commutation relations (CCR)

$$\displaystyle{ [\Pi (\vec{y}),\varphi (\vec{x}))] = \frac{1} {i} \delta (\vec{x} -\vec{ y})\text{,}\quad \vec{x},\vec{y} \in \mathbb{R}^{d-1}\text{,} }$$
(13)

-2pc]Please provide opening parenthesis for “\([\Pi (\vec{y}),\varphi (\vec{x}))]\)” on some suitable dense domain in some Hilbert space.

Since the situation for the case \(v\not\equiv 0\) is quite complicated, because of the well known problem of divergences, let us first consider the case where v ≡ 0, i.e. the case of free fields describing first Level 0–2 for this case. If v ≡ 0 in S Q F as given by (4), then the natural Hilbert space for the CCR representation is \(L^{2}(\mathcal{S}'(\mathbb{R}^{d-1}),\mu _{E}^{0,0})\), with \(\mu _{E}^{0,0} = N(0;(-\Delta _{d-1} + 1)^{-\frac{1} {2} })\), the free time zero field measure (of mass 1), which can be realized on \(\mathcal{S}'(\mathbb{R}^{d-1})\). One shows that \(\mu _{E}^{0,0}\) is the restriction to the σ-algebra generated by the time zero fields of the measure \(\mu _{E}^{0}\), described heuristically as \(\mu _{E}^{0} = Z^{-1}e^{-S_{E}^{0}(\gamma ) }\,\mathrm{d}\gamma\) with

$$\displaystyle{ S^{0}(\gamma ):= \frac{1} {2}\int _{\mathbb{R}}\int _{\mathbb{R}^{d-1}}\left (\left \vert \dot{\gamma }(s,\vec{x})\right \vert ^{2} + \left \vert \nabla \gamma (s,\vec{x})\right \vert ^{2} + \left \vert \gamma (s,\vec{x})\right \vert ^{2}\right )\,\mathrm{d}s\,\mathrm{d}\vec{x}. }$$
(14)

\(\mu _{E}^{0}\) is realized rigorously as \(N(0;(-\Delta _{d} + 1)^{-1})\) on \(\mathcal{S}'(\mathbb{R}^{d})\). It is called the Euclidean free field measure, see [98].

The above \(L^{2}(\mathcal{S}'(\mathbb{R}^{d-1}),\mu _{E}^{0,0})\) is naturally isomorphic to the Fock space (“second quantization”), see [78].

Consider the classical Dirichlet form \(\mathcal{E}(u,v) = \frac{1} {2}\int \nabla u\nabla v\,\mathrm{d}\mu _{E}^{0,0}\), so that \(\mathcal{E}(u,v) = \left (u,-L_{\mu _{E}^{0,0}}v\right )\) for u, v in the relevant domains, in \(L^{2}\left (\mu _{E}^{0,0}\right ):= L^{2}(\mathcal{S}'(\mathbb{R}^{d-1}),\mu _{E}^{0,0})\) with \(L_{\mu _{E}^{0,0}}\) the associated Dirichlet operator. One has that \(H^{0}:= -L_{\mu _{E}^{0,0}}\) is self-adjoint (on its natural domain) and is a realization of the Hamiltonian for the free relativistic quantum field. In an analogous way one can consider the generators of the whole Poincaré group on space-time \(\mathbb{R} \times \mathbb{R}^{d-1}\).

The generators of the Lorentz group are not positive, but can be shown to be essentially self-adjoint on smooth cylinder functions (by using the isomorphism of \(L^{2}(\mu _{E}^{0,0})\) with Fock space, i.e. second quantization).

Then one has a rigorous full implementation of canonical relativistic free quantum fields in \(L^{2}(\mu _{E}^{0,0})\), see [48].

Note that \(L_{\mu _{E}^{0,0}}\) generates a diffusion process \(X_{t}(\vec{x})\), t ≥ 0, \(\vec{x} \in \mathbb{R}^{d-1}\) with state space \(\mathcal{S}'(\mathbb{R}^{d-1})\) satisfying the stochastic differential equation

$$\displaystyle{ \mathrm{d}X_{t}(\vec{x}) = -\left [(-\Delta _{d-1} + 1)^{\frac{1} {2} }X_{t}(\vec{x})\right ]\,\mathrm{d}t +\mathrm{ d}w_{t}(\vec{x})\text{,}\quad t \geq 0\text{,}\ \vec{x} \in \mathbb{R}^{d-1}. }$$
(15)

\(\mathrm{d}w_{t}(\vec{x})\) denotes the Gaussian white noise in the t- and \(\vec{x}\)-variables.

This was first discussed in [23] and [81]. Note also that \(\mu _{E}^{0,0}\) is the unique stationary measure for \(X_{t}(\vec{x})\).

The path space distribution of \(X_{t}(\vec{x})\) is the Euclidean free field measure \(\mu _{E}^{0} = N\left (0;(-\Delta _{\mathbb{R}^{d}} + 1)^{-1}\right )\) on \(\mathcal{S}'(\mathbb{R}^{d})\), as described above.

This has been discussed originally by K. Symanzik [112] and E. Nelson [98] (and, in connection with statistics, by L. Pitt, [103]).

That the restriction of \(\mu _{E}^{0}\) to the σ-algebra σ(X 0) generated by the time zero fields X 0( ⋅ ) can be identified with \(\mu _{E}^{0,0}\) due to the global Markov property of the Euclidean free field [29, 38, 98, 104].

The Level 2 is described by the SDE (SPDE)

$$\displaystyle{ \mathrm{d}X_{\tau }(y) = (\Delta _{d} - 1)X_{\tau }(y) +\mathrm{ d}W_{\tau }(y)\text{,} }$$
(16)

\(y \in \mathbb{R}^{d}\), τ ≥ 0 (τ is a “computer time”, y the space-time variable). \(\mathrm{d}W_{\tau }(y)\) denotes the Gaussian white noise in the τ- and y-variables.

The invariant measure to X τ (y) is \(\mu _{E}^{0}\). The SPDE (16) is called stochastic quantisation equation (SQE) associated with the Euclidean free field (over \(\mathbb{R}^{d}\)).

Already for d = 2, in the case \(v\not\equiv 0\) the construction of Levels 0–2 is much more complicated.

Levels 0 and 1 have been achieved first in models with exponential interaction (for which v is of the form v(u) = e α u, α < α 0, for suitable α 0 > 0, \(u \in \mathbb{R}\)) with renormalization, in [82] resp. [20] (see also [11, 21, 89]).

In the same period the case of v of polynomial type, with renormalization, was achieved, see [71, 106], and references therein.

The model with v of trigonometric type was first discussed in [19] (with regularization, see also [14]) and solved (with renormalizations) in [68] (see also [14, 67]).

All models are covered and unified by the white noise calculus approach [16–18].

Let us give a short summary of these constructions. At Level 0 one constructs a non Gaussian μ v, 0 on \(\mathcal{S}'(\mathbb{R}^{d-1})\). At Level 1 one constructs the analogue of (12) for quantum fields, i.e.

$$\displaystyle{ \mu _{E}^{v}(\mathrm{d}\gamma ) = \textquotedblleft Z^{-1}e^{-S(\gamma )}\mathrm{d}\gamma ''\text{,} }$$
(17)

with \(S(\gamma ) = S^{0}(\gamma ) +\iint v(\gamma (s,\vec{x}))\,\mathrm{d}s\,\mathrm{d}\vec{x}\). (Thus \(\mu _{E}^{v}\) coincides with the expression given by (5) and (6)). At Level 2 the measure \(\mu _{E}^{v}\) appears as invariant measure for the solution of the stochastic quantization equation (also called, in other contexts, Allen-Cahn equation, see, e.g., [99], also related to the Ginzburg-Landau equation), associated to Euclidean fields with interaction given by v:

$$\displaystyle{ \mathrm{d}X_{\tau }(y) = (\Delta _{d} - 1)X_{\tau }(y)\,\mathrm{d}\tau - v'(X_{\tau }(y))\,\mathrm{d}\tau +\mathrm{ d}w_{\tau }(y)\text{,}\quad \tau \geq 0,\ y \in \mathbb{R}^{d}. }$$
(18)

The associated classical Dirichlet forms given by the restriction \(\mu _{E}^{v,0}\) of \(\mu _{E}^{v}\) to the σ-algebra of the zero fields, which is thus a measure on \(S'(\mathbb{R}^{d-1})\), (coinciding with the level 0 measure μ v, 0) resp. by \(\mu _{E}^{v}\) itself, yield in turn generators of Markov processes (diffusions) \(X_{t}(\vec{x})\) resp. \(X_{\tau }(y), t,\tau \geq 0\), \(\vec{x} \in \mathbb{R}^{d-1}\), \(y \in \mathbb{R}^{d}\), the latter solving (18).

The generator of \(X_{t}(\vec{x})\) is identifiable (via a natural isomorphism) on smooth cylinder functions with the Hamiltonian acting in \(L^{2}\left (\mu _{E}^{v,0}\right )\), of the interacting quantum fields, given the global Markov property of the Euclidean fields [29]. The generators of the Poincaré group are also realizable as self-adjoint operators on \(L^{2}(\mu _{E}^{v,0})\) [21, 47].

Whether all generators are already determined on smooth cylinder functions is a hard problem, only solved for the case X τ (y) in a bounded region when d = 2 [61, 90] (it has been solved in bounded and unbounded regions for d = 1 in [31, 32, 87]).

For recent work on Level 2 for d = 2 with exponential interaction see [30, 31].

Remark 2

The SQE has also been studied intensively with noise regularized in the y-variable and corresponding modified drift coefficient, so as to heuristically maintain the same invariant measure.

See [41–43, 56, 60, 61, 64, 70, 83–86, 95–97, 99, 100, 105], and also e.g. [58] for related problems.

For d = 3 partial results on Level 0–2 have been achieved for v which are of the form of a 4-th power (with renormalization), see, e.g. the references in [36].

Recently solutions of the SQE of Level 2 has been constructed by different methods in [77]. However the Markov character of the solutions (a property included if one uses processes associated with Dirichlet forms) has not yet been discussed.

The difficulties in constructing solutions is related to the expected singularity of \(\mu _{E}^{v}\) with respect to \(\mu _{E}^{0}\) even in bounded regions (this has been proved rigorously in [35] only for the restriction of these measure to the σ-algebra generated by the time zero fields).

Other types of relativistic models however also exist for all d including d = 4, provided one relaxes the axioms to the ones which have been introduced (by Strocchi-Wightman and Morchio-Strocchi) for gauge fields, see the work [73] (in white noise analysis) and [12, 13] (and references therein).

For the construction of gauge fields for d = 2 see [1, 24, 25, 47, 48] (and references therein).

For d = 3 in the Euclidean case a model of gauge fields has been constructed using μ F (Chern-Simons model) [15, 47, 91], again using methods from white noise analysis (to which Ludwig has fundamentally contributed).

5 Some Remarks on Related Systems

Methods related to the ones discussed in the preceding Sections can also be used in other areas.

E.g., in recent years equations for neuronal dynamics of the FitzHugh-Nagumo type have been discussed in [5, 6].

In the simplest version they describe a signal propagating along a single neuron under the influence of external noise and of certain salts concentrations. They are of the form of 2 coupled random variables, the components of a vector X ∈ L 2([0, 1]) × L 2([0, 1]) satisfying a SDE of the form dX t  = A X t  dt + F(X t ) dt +ɛ dw t , t ≥ 0, with w t space-time Gaussian white noise and ɛ > 0 a (small diffusion) parameter.

A is a diffusion operator, F(x) is a non linear term, e.g., of the Fitz-Hugh-Nagumo type \(F(x) = \left (\begin{array}{*{10}c} -x(x - 1)(x-\theta ),\quad 0 <\theta < 1\\ 0 \end{array} \right ),x \in \mathbb{R}\).

Using dissipativity, in [5, 54] the existence, uniqueness of solutions and the uniqueness of the invariant measure have been discussed, as well as the \(\varepsilon \downarrow 0\) behaviour in terms of detailed asymptotic expansions, with L p-control, 1 ≤ p < +∞, on remainders.

The invariant measure is exhibited, as being of Gibbsian type with a density e −G with respect to μ A , and G′ = −F, where μ A is the invariant Gaussian measure associated to the linear stochastic equation for F ≡ 0.

It is interesting that in the case of Gaussian white noise replaced by Lévy noise also an invariant measure has been found in [6]. Its character is presently under discussion.

In the finite dimensional case explicit invariant measures of related systems can be constructed via an analogue of the general ground state transformation discussed early in [27], see [6, 50]. Andrisani and Cufaro-Petroni [50] have interesting applications, e.g. to the study of halo formation in intense beams of charged particles in accelerators.

Stochastic pde’s of the form of those we have discussed in connection with quantum field theory and neuronal dynamics also occur in many other areas, including, e.g., hydrodynamics and polymer physics, to which Ludwig Streit has given outstanding contributions. Let us mention, in particular, his work on the Burgers equation [51] and his work on polymer-type measures, a prototype for the latter being given by the Edwards’ model described heuristically by the invariant measure

$$\displaystyle{ \mu (\mathrm{d}\gamma ) = \text{\textquotedblleft }Z^{-1}e^{-\lambda \int _{0}^{t}\int _{ 0}^{t}\delta (\omega (s)-\omega (s'))\,\mathrm{d}s\,\mathrm{d}s' }\mu _{0}(\mathrm{d}\omega )\text{\textquotedblright } }$$
(19)

μ 0 being Wiener measure for Brownian motion on \(\mathbb{R}^{d}\), δ the Dirac distribution, λ a (positive) constant, t > 0. The Gibbs factor inhibits self-intersection of paths. For such measures and related ones see, e.g., [9, 40, 62, 63].

Varadhan showed in [113] that setting \(L_{\varepsilon } =\int _{ 0}^{t}\int _{0}^{t}\delta _{\varepsilon }(\omega (s) -\omega (s'))\,\mathrm{d}s\,\mathrm{d}s'\) with δ ɛ a natural ɛ-regularization of δ one has that L ɛ − E(L ɛ ) converges as ɛ ↓ 0 in \(L^{2}(\mu _{0})\) to a limit \(\tilde{L}\), in such a way that

$$\displaystyle{ \|\tilde{L}-\left [L_{\varepsilon } - E(L_{\varepsilon })\right ]\|_{L^{2}(\mu _{0})} \leq C\varepsilon ^{\alpha } }$$
(20)

for some constant C > 0, for all \(\alpha < \frac{1} {2}\).

This implies in particular the existence of the polymer measure μ, understood as weak limit of μ ɛ as ɛ ↓ 0 defined by (19) with

$$\displaystyle{ L:=\int _{ 0}^{t}\int _{ 0}^{t}\delta (\omega (s) -\omega (s'))\,\mathrm{d}s\,\mathrm{d}s' }$$
(21)

replaced by L ɛ − E(L ɛ ).

The above estimate (20) has been improved to α < 1 in a recent publication by Ludwig and coworkers [53].

6 Conclusions

We have just been able to discuss a few of the many important developments inspired, initiated and rigorously pursued by Ludwig Streit. I am very grateful to Ludwig for having been a constant source of inspiration for me and many coworkers since many years. I wish him many more years of good health, happiness, “frohes Schaffen”.