Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

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

$\displaystyle \left\Vert {{H^{ - 1/2}}({{({\text{ad}}\;A)}^n}H){H^{ - 1/2}}u} \right\Vert \leq {c_n}\left\Vert u \right\Vert$

instead of the norm estimates

$\displaystyle \left\Vert {{{({\text{ad}}\;A)}^n}Hu} \right\Vert \leq {c_n}\left\Vert {Hu} \right\Vert$

usually required for this type of theorem. We call the new estimates ``quadratic form type", since they are sometimes equivalent to

$\displaystyle \vert({({\text{ad}}\;A)^n}Hu,u)\vert \leq {c_n}\vert(Hu,u)\vert.$

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.

References [Enhancements On Off] (What's this?)

  • [1] J. M. Cook, The mathematics of second quantization, Trans. Amer. Math. Soc. 74 (1953), 222-245. MR 14, 825. MR 0053784 (14:825h)
  • [2] W. G. Faris, Invariant cones and uniqueness of the ground state for fermion systems, J. Math. Phys. 13 (1972), 1285-1290. MR 0321451 (47:9984)
  • [3] J. Glimm, Boson fields with nonlinear selfinteraction in two dimensions, Comm. Math. Phys. 8 (1968), 12-25. MR 0231601 (37:7154)
  • [4] J. Glimm and A. Jaffe, A $ \lambda {\phi ^4}$ quantum field theory without cut-offs. I, Phys. Rev. (2) 176 (1968), 1945-1951. MR 40 #1106. MR 0247845 (40:1106)
  • [5] -, The $ \lambda {({\phi ^4})_2}$ quantum field theory without cut-offs. II. The field operators and the approximate vacuum, Ann. of Math. (2) 91 (1970), 362-401. MR 41 # 1333. MR 0256677 (41:1333)
  • [6] R. Goodman, Analytic domination by fractional powers of a positive operator, J. Functional Analysis 3 (1969), 246-264. MR 39 #801. MR 0239444 (39:801)
  • [7] L. Gross, Analytic vectors for representations of the canonical commutation relations and non-degeneracy of ground states, Cornell University preprint, November, 1972.
  • [8] -, Existence and uniqueness of physical ground states, J. Functional Analysis 10 (1972), 52-109. MR 0339722 (49:4479)
  • [9] T. Kato, Perturbation theory for linear operators, Die Grundlehren der math. Wissenschaften, Band 132, Springer-Verlag, New York, 1966. MR 34 #3324. MR 0203473 (34:3324)
  • [10] E. Nelson, Analytic vectors, Ann. of Math. (2) 70 (1959), 572-615. MR 21 #5901. MR 0107176 (21:5901)
  • [11] -, A quartic interaction in two dimensions, Math. Theory of Elementary Particles (Proc. Conf., Dedham, Mass., 1965), M. I. T. Press, Cambridge, Mass., 1966, pp. 69-73. MR 35 # 1309. MR 0210416 (35:1309)
  • [12] -, Schrödinger particles interacting with a quantized scalar field, Proc. Conf. on Theory and Appl. of Analysis in Function Space (Dedham, Mass., 1963), M. I. T. Press, Cambridge, Mass., 1964, pp. 87-120. MR 35 #2590. MR 0167152 (29:4425)
  • [13] L. Rosen, The $ {({\phi ^{2n}})_2}$ quantum field theory: Higher order estimates, Comm. Pure Appl. Math. 24 (1971), 417-457. MR 44 #5042. MR 0287840 (44:5042)
  • [14] I. Segal, Tensor algebras over Hilbert spaces. I, Trans. Amer. Math. Soc. 81 (1956), 106-134. MR 17, 880. MR 0076317 (17:880d)
  • [15] B. Simon, Ergodic semigroups of positivity preserving self-adjoint operators (preprint, 1972). MR 0358434 (50:10900)
  • [16] B. Simon and R. Høegh-Krohn, Hpercontractive semigroups and two dimensional self-coupled Bose fields, J. Functional Analysis 9 (1972), 121-180. MR 45 #2528. MR 0293451 (45:2528)
  • [17] A. Sloan, A non perturbative approach to non degeneracy of ground states in quantum field theory: Polaron models, Georgia Tech. (1973) (preprint).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 81.47

Retrieve articles in all journals with MSC: 81.47


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

American Mathematical Society