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The method of dimensional regularization has proved itself very useful especially when it is applied to the gauge theories,3 The applications have been limited mainly to the calculations of Feynman integrals. We want to apply the method also to the calculation of spectral representations. This was motivated by our previous attempt to remove the difficulty of the critical dimension in the dual string model.4

As one can see at this conference, and similarly as with some other concepts of theoretical physics, the Feynman integral is being used as an important tool of quantum dynamics at a time when its mathematical formalization is still being developed, — a novel and remarkable example of such development was presented by Mme Sirugue [1].

In the Feynman formulation of quantum mechanics the formal expression for the solution of Schrodinger’s equation in L2 (ℝd) $$ (f,g) = \int {g \wedge {f^{ + }}{v^{'}}} $$ $$ \psi (o,X) = \varphi (X) $$ is given by $$ \psi (t,X) = ({e^{{\frac{i}{h}tH}}}\varphi )(X) = N\int {_{{y(t) = X}}{e^{{\frac{i}{h}{S_{t}}(\gamma )}}}} \varphi (\gamma (o))dy $$ (1.1) where H = Ho + V, Ho being the free energy operator \( {H_{o}} = - \frac{{{h^{2}}}}{{2m}}\Delta \) and Δ being the Laplacian in ℝd. The classical action St (γ) is equal to $$ {S_{t}}(y) = \int\limits_{o}^{t} {\frac{m}{2}\mathop{{{\gamma ^{2}}}}\limits^{ \bullet } (\tau )d\tau - \int\limits_{o}^{t} V } (\gamma (\tau ))d\tau $$ (1.2)