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

MSC:

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

Zbl 0649.00017