A Modified Garding-Wightman Axioms with a Modification on the Symmetry of Field Operators, an Explicit Formulation by Means of Complex Valued Random Fields Defined through General Lévy Fields on $\mathbb{R}^d $
A talk in the Bielefeld Stochastic Afternoon series by
Yoshida Minoru
| Abstract: | A modification of Hermitian scalar quantum fields framework, in the sense of the Garding-Wightman Axioms, is considered. The modified quantum fields \( < \mathcal{H}, U, \psi, D > \) constructed here satisfy all the requirements of the Garding-Wightman Axioms, except that the field operators \( \psi(f) \) with \( f \in \mathcal{S}(\mathbb{R}^d \to \mathbb{R}) \) are symmetric operators on the physical Hilbert space \( \mathcal{H} \). Namely, the field operators \( \psi^{+}(f) \) and \( \psi^{-}(f) \), which are complex conjugate to each other in some sense, defined here are not symmetric operators on \( \mathcal{H} \) for \( f \in \mathcal{S}(\mathbb{R}^d \to \mathbb{R}) \). However, e.g., for \( f \in \mathcal{S}(\mathbb{R}^d \to \mathbb{R}) \), that is a real valued even function, the combination of the operators \( \psi^{+}(f) + \psi^{-}(f) \) defines a symmetric operator on \( \mathcal{H} \), which naturally corresponds to a composition of creation and annihilation operators (see (iv) in Theorem 1). The present construction of \( < \mathcal{H}, U, \psi, D > \) (denoting \( \psi = \psi^{+} \) or \( \psi^{-} \)) is relatively simple and direct, not passing through the Euclidean strategies, e.g., the Osterwalder-Schwinger Axioms or Nelson’s Axioms. To define \( \psi^{+}(f) \) and \( \psi^{-}(f) \) on \( \mathcal{H} \), general L\'evy fields on \( \mathbb{R}^d \) are used. More precisely, \( \psi^{+}(f) \) and \( \psi^{-}(f) \) are, respectively, defined through the Fourier and Fourier inverse transform of \[ j_{\gamma}P^{+}(\tau, \xi_1, \ldots, \xi_{d-1}) = \begin{cases} (\tau^2 - ( \sum_{k=1}^{d-1} \xi_k^2 + m^2 ))^{-\gamma}, & \tau > \sqrt{ \sum_{k=1}^{d-1} \xi_k^2 + m^2 } \\ 0, & \text{otherwise}, \end{cases} \] for \( \gamma \in (0, \frac{1}{2}] \) (\( P^{+} \) denotes the restricted Poincare group), accompanied with the Levy or the centered Gaussian fields \( \mu \) on the real distribution space \( \mathcal{S}'(\mathbb{R}^d \to \mathbb{R}) \), of which characteristic functions are (in the sense of the Bochner-Minlos theorem) \[ \int_{\mathcal{S}'(\mathbb{R}^d \to \mathbb{R})} e^{i \langle g, \varphi \rangle} d\mu(\varphi) = e^{- \int_{\mathbb{R}^d} \left( \int_{\mathbb{R}\setminus\{0\}} ( e^{i s g(x)} - 1 ) M(ds) \right) dx }, \] and \[ \int_{\mathcal{S}'(\mathbb{R}^d \to \mathbb{R})} e^{i \langle g, \varphi \rangle} d\mu(\varphi) = e^{- \int_{\mathbb{R}^d} |g(x)|^2 dx }, \quad \text{for real } g \in \mathcal{S}(\mathbb{R}^d \to \mathbb{R}). \] Keywords: Axiomatic quantum field theory, Garding-Wightman axioms, Bochner-Minlos theorem, Levy fields on \( \mathbb{R}^d \). MSC (2020): 31C25, 46E27, 46N30, 46N50, 47D07, 60H15, 60J46, 60J75, 81S20. References [A,H,W 1996] Albeverio, S., Gottschalk, H., Wu, J.L., Convoluted generalized white noise, Schwinger functions and their analytic continuations to Wightman functions. Rev. Math. Phys. 8 (1996), 763--817. [A,H,W 1997] Albeverio, S., Gottschalk, H., Wu, J.-L.: Models of local relativistic quantum fields with indefinite metric (in all dimensions). Commun. Math. Phys. 184 (1997), 509--531. [Hida] Hida, T.: Brownian motion. [R,S] Reed, M., Simon, B.: Fourier Analysis, Self-Adjointness, Academic Press. [S] Simon, B.: The \( P(\Phi)^2 \) Euclidean (Quantum) Field Theory, Princeton. [Ito S.] Seizo Ito: Functional analysis III, Iwanami Kiso Suugaku. Within the CRC this talk is associated to the project(s): B1 |