Skip to main content

Critical Exponents and Renormalization in the (⌽)4 Scaling Limit

  • Conference paper
Quantum Dynamics: Models and Mathematics

Part of the book series: Acta Physica Austriaca ((FEWBODY,volume 16/1976))

Abstract

For dimensions d ≤ 3, the ⌽4 scaling limit defines a nonrenormalizable field theory. The standard relations between critical exponents and renormalization are presented. Arguments supporting the existence of the scaling limit are based on correlation inequalities and the numerical values of Ising model exponents, \( 2\eta _ \ne ^ < \eta _E \) for d=2,3.

Supported in part by the National Science Foundation under Grant 74-13252.

Supported in part by the National Science Foundation under Grant 75-21212.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Baker, Self-interacting boson quantum field theory and the thermodynamic limit in d dimensions, J. Math. Phys. 16, 1324–1346 (1975).

    Article  ADS  Google Scholar 

  2. E. Barouch, B. Mccoy, C. Tray and T. T. WU, The spin-spin correlation functions for the two dimensional Ising-model Exact theory in the scaling limit, to appear.

    Google Scholar 

  3. J. Feldman, On the absence of bound states in the ⌽4 quantum field model without symmetry breaking, Cand. J. Phys. 52, 1583–1587 (1974).

    ADS  Google Scholar 

  4. J. Feldman and K. Osterwalder, The Wightman axioms and the mass gap for weakly coupled (\( \left( \varphi \right)_3^4 \)) quantum field theories, to appear.

    Google Scholar 

  5. M. Fisher, Rigorous inequalities for critical point correlation exponents, Phys. Rev. 180, 594–600 (1969).

    Article  ADS  Google Scholar 

  6. J. Fröhlich, Existence and analyticity in the bare parameters of the \( \left| {\lambda \left( {\overrightarrow \varphi \overrightarrow \varphi } \right)^2 - \sigma \varphi _1^2 - \mu \varphi _1 } \right| \) quantum field models, I. Manuscript.

    Google Scholar 

  7. J. Glimm and A. Jaffe, The (λ\( \left( {\lambda \varphi ^4 } \right)_2 \) quantum field theory without cutoffs III. The physical vacuum, Acta Math. 125, 203–261 (1970).

    Article  MathSciNet  Google Scholar 

  8. J. Glimm and A. Jaffe, The (λ\( \left( {\lambda \varphi ^4 } \right)_2 \) quantum field theory without cutoffs IV. Perturbation of the Hamiltonian, J. Math. Phys. 13, 1558–1584 (1972).

    Article  MathSciNet  ADS  Google Scholar 

  9. J. Glimm and A. Jaffe, \( \varphi _2^4 \) quantum field theory in the single phase re-gion: Differentiability of the mass and bounds on critical exponents, Phys. Rev. D 10, 536–539 (1974).

    Article  ADS  Google Scholar 

  10. J. Glimm and A. Jaffe, Two and three body equations in quantum field models, Commun. Math. Phys. 44, 293–320 (1974).

    Article  MathSciNet  ADS  Google Scholar 

  11. J. Glimm and A. Jaffe, Remark on the existence of \( \varphi _4^4 \). Phys. Rev. Lett. 33, 440–442 (1974).

    Article  MathSciNet  ADS  Google Scholar 

  12. J. Glimm and A. Jaffe, Three particle structure of ⌽4 interactions and the scaling limit, Phys. Rev. D 11, 2816–2827 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  13. J. Glimm, A. Jaffe, and T. Spencer, The particle structure of the weakly coupled P(⌽)2 models and other applications of high temperature expansions. In: Constructive quantum field theory, G. Velo and A. Wightman (eds.) Springer Verlag, Berlin, 1973.

    Google Scholar 

  14. J. Glimm, A. Jaffe, and T. Spencer, Existence of phase transitions for \( \varphi _2^4 \) quantum fields, Commun. Math. Phys. To appear.

    Google Scholar 

  15. J. Glimm, A. Jaffe, and T. Spencer, A cluster expansion for the \( \varphi _2^4 \) quantum field theory in the two phase region. In preparation.

    Google Scholar 

  16. F. Guerra, L. Rosen, and B. Simon, In: Constructive quantum field theory, G. Velo and A. Wightman (eds.) Springer Verlag, Berlin, 1973.

    Google Scholar 

  17. D. Isaacson, The critical behavior of the autoharmonic oscillator, NYU Thesis.

    Google Scholar 

  18. B. Josephson, Inequality for the specific heat, I Derivation, II Applications, Proc. Phil. Soc. 92, 269–284 (1967).

    Article  ADS  Google Scholar 

  19. D. Marchesin, Work in progress.

    Google Scholar 

  20. J. Magnen and R. Seneor, The infinite volume limit of the \( \varphi _3^4 \) model, Ann. Inst. H. Poincaré, to appear.

    Google Scholar 

  21. K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Commun. Math. Phys. 31 83–112 (1974).

    Article  MathSciNet  ADS  Google Scholar 

  22. K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions II, Commun. Math. Phys. 42, 281–305 (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Y. Park, Uniform bounds of the pressure of the λ\( \lambda \varphi _3^4 \) field model. Preprint.

    Google Scholar 

  24. J. Rosen, Mass renormalization for λ\( \lambda \varphi _2^4 \) Euclidean lattice field theory.

    Google Scholar 

  25. J. Rosen, Private communication.

    Google Scholar 

  26. E. Seiler and B. Simon, Nelson’s symmetry and all that in the Yukawa and \( \varphi _3^4 \) field theories. Preprint.

    Google Scholar 

  27. T. Spencer, The absence of even bound states in \( \varphi _2^4 \cdot \). Commun. Math. Phys. 39, 77–79 (1974).

    Article  ADS  Google Scholar 

  28. E. Brezin, J. C. Leguillore and J. Zinn-Justin, Field theoretical approach to critical phenomena. In: Phase transitions and critical phenomena, Vol. VI., Ed. by Domb and Green, Academic Press, New York, to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1976 Springer-Verlag

About this paper

Cite this paper

Glimm, J., Jaffe, A. (1976). Critical Exponents and Renormalization in the (⌽)4 Scaling Limit. In: Streit, L. (eds) Quantum Dynamics: Models and Mathematics. Acta Physica Austriaca, vol 16/1976. Springer, Vienna. https://doi.org/10.1007/978-3-7091-8473-8_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-8473-8_9

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-8475-2

  • Online ISBN: 978-3-7091-8473-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics