Constructions of measures and quantum field theories on mapping spaces. (English) Zbl 1097.58007

Taubes, Clifford H.

J. Differ. Geom. 70, No. 1, 23-57 (2005).

A simple construction of measures on \(\text{Map}(M, X)= X^M\), \(M\) a topological space and \(X\) a compact oriented Riemannian manifold, is presented.
The construction uses the heat kernel measure associated to the differential operator \[ A_z= \sum_{1\leq j\leq N} a(z_i,z_j)\sum_{1\leq\alpha\leq d}\partial^i_\alpha \partial^j_\alpha,\quad d=\dim X, \] where \(\{\partial_1,\dots,\partial_d\}\) generate \(TP/\ker(\pi_*)\), \(\pi: P\to X\) is a compact fiber bundle with the volume form \(dp\) and having such set of vector fields, and \(a: M\times M\to[0,\infty)\) be a continuous function. The heat equation on \(\overbrace{P\times\cdots\times P}^{N}= \times_N P\) has a measure valued solution \(K_z\); \[ \begin{aligned} {d\over ds} \int_{\times_N P} FK_z|_s\,dp &= \int_{\times_N P} (A_zF)K_z|_s \,dp,\\ \int_{\times_N P}FK_z|_{s=0}\,dp &= \int_{\times_N P}F\delta\,N\,dp,\end{aligned} \] where \(F\) is a twice differentiable function. It is known that \(K_z\) is smooth for \(s> 0\) if \(\sum_{1\leq i\leq j\leq N} a(z_i, z_j)\eta_i\eta_j\) is non-degenerated [L. Hörmander, Proc. Symp. Pure Math. 10, 138–183 (1967; Zbl 0167.09603), hereafter referred to as [1]]. If \(\{\partial_1,\dots, \partial_d\}\) are the Lie algebra generator of \(TP\), then

1.

\(K_z\geq 0\),

2.

\(\int_P K_{(z_1,\dots, z_N)}(p_1,\dots, p_{N-1},p)\,dp= K_{(z_1,\dots, z_{N-1})}(p_1,\dots, p_{N-1})\),

3.

\(K_z(p_1,\dots, p_N)= \delta^{N-N'+ 1}(p_{N'},\dots, p_N)(K_{(z_1,\dots, z_{N'-1},z_{N'})}(p_1,\dots, p_{N'})\),

4.

If \(\sigma\) is a permutation of \(\{1,\dots, N\}\), then \(K_z= \sigma^*(K_{\sigma(z)})\), ((1.6) and Lemma 2.1)).

These are proved in §2 using results in [1] and in [D. W. Stroock and S. R. S. Varadhan, Multidimensional diffusion processes, Grundlehren der mathematischen Wissenschaften. 233. Berlin, Heidelberg, New York: Springer-Verlag (1979; Zbl 0426.60069), hereafter refered to as [2]].
Labelling the generators of the \(\sigma\)-algebra generated by the cylinder sets in \(P^M\) as \(\{(z_i, U_i)\}_{1\leq i\leq N}\), \(z= (z_1,\dots, z_N)\in\times_N M\), each \(U_j\) is an open subset of \(P\), the measure of this set is defined to be \[ \int_{\times_{1\leq j\leq N}U_j} K_z|_{s=1} \,dp. \] By (1.6) and the Kolmogorov extension theorem (cf.[2]), a measure on \(P^M\) is obtained (Lemma 2.3). It induces a measure on \(X^M\). As for the support of this measure, it is shown if \(a\) is Hörmander continuous for some positive exponent, then it induces a measure on the space of continuous maps from \(M\) to \(P\) (Theorem 2.4). In §3, Gaussian measure is reviewed: In general, these two measures are distinct, but the same if \(X\) has a flat metric. These measures are applied to construct Hamiltonian quantum field theory (Def. 4.1) and show if \(M= \mathbb{R}\times Y\), \(Y\) is a compact Riemannian manifold, and \(a\) is a regular Green’s function (Th. 4.4, proof of this theorem is given in §5). The author remarks even on \(S^1\), no second-order differential operator has a regular Green’s function. So some renormalization is needed. The author says this will be discussed in the sequal of this paper. §6, the last §, additional map \(\theta: M\to P\) is introduced and measures with parameters \(\theta\) are considered.

Reviewer: Akira Asada (Takarazuka)

MSC:

58D20	Measures (Gaussian, cylindrical, etc.) on manifolds of maps
81T08	Constructive quantum field theory
58D15	Manifolds of mappings

Keywords:

space of maps; heat kernel measure; Gaussian measure; Hamiltonian quantum field theory

Citations:

Zbl 0167.09603; Zbl 0426.60069