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

MathSciNet review:
0345564

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Abstract | References | Similar Articles | Additional Information

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

*H*the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and , the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as where

*dq*is a probability measure on

*Q*, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then

*maximizes support*in the sense that is nonzero almost everywhere whenever

*f*is not identically zero.

**[1]**J. M. Cook,*The mathematics of second quantization*, Trans. Amer. Math. Soc.**74**(1953), 222–245. MR**0053784**, 10.1090/S0002-9947-1953-0053784-4**[2]**William G. Faris,*Invariant cones and uniqueness of the ground state for fermion systems*, J. Mathematical Phys.**13**(1972), 1285–1290. MR**0321451****[3]**James Glimm,*Boson fields with the :Φ⁴: interaction in three dimensions*, Comm. Math. Phys.**10**(1968), 1–47. MR**0231601****[4]**James Glimm and Arthur Jaffe,*A 𝜆𝜑⁴ quantum field without cutoffs. I*, Phys. Rev. (2)**176**(1968), 1945–1951. MR**0247845****[5]**James Glimm and Arthur Jaffe,*The 𝜆(Π⁴)₂ quantum field theory without cutoffs. II. The field operators and the approximate vacuum*, Ann. of Math. (2)**91**(1970), 362–401. MR**0256677****[6]**Roe Goodman,*Analytic domination by fractional powers of a positive operator*, J. Functional Analysis**3**(1969), 246–264. MR**0239444****[7]**L. Gross,*Analytic vectors for representations of the canonical commutation relations and non-degeneracy of ground states*, Cornell University preprint, November, 1972.**[8]**Leonard Gross,*Existence and uniqueness of physical ground states*, J. Functional Analysis**10**(1972), 52–109. MR**0339722****[9]**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****[10]**Edward Nelson,*Analytic vectors*, Ann. of Math. (2)**70**(1959), 572–615. MR**0107176****[11]**Edward Nelson,*A quartic interaction in two dimensions*, Mathematical Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965) M.I.T. Press, Cambridge, Mass., 1966, pp. 69–73. MR**0210416****[12]**Edward Nelson,*Schrödinger particles interacting with a quantized scalar field*, Analysis in function space, M.I.T. Press, Cambridge, Mass., 1964, pp. 87–120. MR**0167152****[13]**Lon Rosen,*The (𝜑²ⁿ)₂ quantum field theory: higher order estimates*, Comm. Pure Appl. Math.**24**(1971), 417–457. MR**0287840****[14]**I. E. Segal,*Tensor algebras over Hilbert spaces. I*, Trans. Amer. Math. Soc.**81**(1956), 106–134. MR**0076317**, 10.1090/S0002-9947-1956-0076317-8**[15]**Barry Simon,*Ergodic semigroups of positivity preserving self-adjoint operators*, J. Functional Analysis**12**(1973), 335–339. MR**0358434****[16]**Barry Simon and Raphael Høegh-Krohn,*Hypercontractive semigroups and two dimensional self-coupled Bose fields*, J. Functional Analysis**9**(1972), 121–180. MR**0293451****[17]**A. Sloan,*A non perturbative approach to non degeneracy of ground states in quantum field theory*:*Polaron models*, Georgia Tech. (1973) (preprint).

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Additional Information

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

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

Article copyright:
© Copyright 1974
American Mathematical Society