A unified approach to classical, bosonic and fermionic Brownian motions. (English) Zbl 0656.60085

Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 303-320 (1988).
[For the entire collection see Zbl 0649.00017.]
Motivated by P. Lévy’s martingale characterization of standard Brownian motion, the canonical commutation and anticommutation relations of quantum fields are derived from suitable martingale hypotheses. Indeed, let \(X=\{X(f),\quad f\in L_ 2({\mathbb{R}}_+)\}\) be a family of selfadjoint operators and \(\phi\) a unit vector in a Hilbert space \({\mathcal H}\) such that the linear manifold \({\mathcal M}\) generated by the set \[ \{\phi,\quad X(f_ 1)...X(f_ n)\phi,\quad f_ j\in L_ 2({\mathbb{R}}_+),\quad 1\leq j\leq n,\quad n=1,2,...\} \] is dense in \({\mathcal H}\). For any \(f\in L_ 2({\mathbb{R}}_+)\) let \(f_{t]}\), \(f_{[t}\) and \(f_{[s,t]}\) denote f \(I_{[0,t]}\), f \(I_{[t,\infty)}\) and f \(I_{[s,t]}\) respectively, for all \(0\leq s<t<\infty\), \(I_ E\) denoting the indicator of E. Let \({\mathcal M}_{t]}\) be the linear manifold generated by \[ \{\phi,\quad X(f_{t]}^{(1)})...X(f_{t]}^{(n)})\phi,\quad f^{(j)}\in L_ 2({\mathbb{R}}_+),\quad 1\leq j\leq n,\quad n=1,2,...\}. \] The family X is called a smooth Lévy field with cyclic vector \(\phi\) and covariance kernel K(f,g), \(f,g\in L_ 2({\mathbb{R}}_+)\) if the following conditions hold:
(i) \(X(0)=0\) and the correspondence \((f_ 1,...,f_ n)\to X(f_ 1)...X(f_ n)\phi\) is real multilinear for each n;
(ii) for fixed \(f^{(j)}\), \(j=1,2,...,n\) the map \(t\to X(f_{t]}^{(1)})...X(f_{t]}^{(n)})\phi\) is continuous in [0,\(\infty];\)
(iii) for \(u,v\in {\mathcal M}_{t]}\), \(t\geq 0\), \(f,g\in L_ 2({\mathbb{R}}_+)\), \[ <u, X(f_{[t})v>=0\quad and\quad <u, X(f_{[t})X(g_{[t})v>=K(f_{[t},\quad g_{[t})<u,v>; \] (iv) \(\{X(f_{t]})X(f_{[t})+\epsilon E X(f_{[t})X(f_{t]})\}u=0\) for all \(u\in {\mathcal M}\), \(f\in L_ 2({\mathbb{R}}_+)\), \(t\geq 0\) where \(\epsilon\) is a constant equal to \(\pm 1;\)
(v) there exist two families of nonnegative Radon measures \(\{\mu_{f,g}\}\), \(\{\nu_{f,g}\}\), \(f,g\in L_ 2({\mathbb{R}}_+)\) such that \[ \| \{X(f_{[s,t]})X(g_{[s,t]})- K(f_{[s,t]},g_{[s,t]})\}u\|^ 2\leq \| u\|^ 2\mu_{f,g}([s,t])\nu_{f,g}([s,t]),\quad u\in {\mathcal M}_{s]}. \] For any such smooth Lévy field the following relations hold: \[ \{X(f)X(g)+\epsilon X(g)X(f)\}=\{K(f,g)+\epsilon K(g,f)\}u \] for all \(u\in {\mathcal M}\), \(f,g\in L_ 2({\mathbb{R}}_+)\).
Reviewer: K.R.Parthasarathy


60J65 Brownian motion
81S05 Commutation relations and statistics as related to quantum mechanics (general)
60H05 Stochastic integrals


Zbl 0649.00017