Analytic domination with quadratic form type estimates and nondegeneracy of ground states in quantum field theory

Author:
Alan D. Sloan

Journal:
Trans. Amer. Math. Soc. **194** (1974), 325-336

MSC:
Primary 81.47

DOI:
https://doi.org/10.1090/S0002-9947-1974-0345564-0

MathSciNet review:
0345564

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Abstract: We present a theorem concerning the analytic domination by a semi-bounded selfadjoint operator *H* of another linear operator *A* which requires only the quadratic form type estimates \[ \left \| {{H^{ - 1/2}}({{({\text {ad}}\;A)}^n}H){H^{ - 1/2}}u} \right \| \leq {c_n}\left \| u \right \|\] instead of the norm estimates \[ \left \| {{{({\text {ad}}\;A)}^n}Hu} \right \| \leq {c_n}\left \| {Hu} \right \|\] usually required for this type of theorem. We call the new estimates “quadratic form type", since they are sometimes equivalent to \[ |({({\text {ad}}\;A)^n}Hu,u)| \leq {c_n}|(Hu,u)|.\] The theorem is then applied with *H* the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and $A = \pi (g)$, the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as ${L^2}(Q,dq)$ where *dq* is a probability measure on *Q*, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then ${e^{ - tH}}$ *maximizes support* in the sense that ${e^{ - tH}}f$ is nonzero almost everywhere whenever *f* is not identically zero.

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Keywords:
Analytic vector,
analytic domination,
quantum fields,
bosons,
semigroup,
positivity preserving

Article copyright:
© Copyright 1974
American Mathematical Society