Equivalence of the path-integral theory of spinning particles and the topological nonlinear <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math></span> model in <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><mo>=</mo><mn>2</mn><mn></mn></math></span> dimensions

Eduardo Fradkin; Enrique Moreno; Fidel A. Schaposnik

doi:10.1103/PhysRevD.45.595

Access by Xinjiang University

Equivalence of the path-integral theory of spinning particles and the topological nonlinear $σ$ model in $d = 2$ dimensions

Eduardo Fradkin and Enrique Moreno

Fidel A. Schaposnik

Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801

Departamento de Fisica, Universidad Nacional de La Plata, C.C. 67, 1900 La Plata, Argentina
Comisión de Investigaciones Cientificas, Buenos Aires, Argentina

PDF

Phys. Rev. D 45, 595 – Published 15 January, 1992

DOI: https://doi.org/10.1103/PhysRevD.45.595

Abstract

We show that the path-integral theory of a spinning particle with spin $S$ is equivalent to the topological O(3) nonlinear $σ$ model (TNLSM) in 1+1 Euclidean dimensions. We prove this equivalence in two different ways. In particular, we use stochastic quantization of the theory of spin to show the identity of the generating functionals of correlation functions of both theories. Also, using canonical quantization of the TNLSM, we show that its ground-state wave function coincides with the Feynman weight for the path-integral theory of the spin. In addition, we give a full classification of the invariants of the TNLSM. The invariants are computed explicitly in the case of the sphere.

References (29)

E. Witten, Commun. Math. Phys. 117, 353 (1988) ibid.118, 411 (1988) ibid.121, 351 (1989)
E. Witten, Phys. Lett. B 206, 601 (1988)
F. D. M. Haldane, Phys. Lett. 93A, 464 (1983)
P. B. Wiegmann, Phys. Rev. Lett. 60, 821 (1988)
E. Fradkin and M. Stone, Phys. Rev. B 38, 7215 (1988)
A. M. Polyakov, in Fields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School, Les Houches, France, 1988, edited by E. Brezin and J. Zinn-Justin, Les Houches Summer School Proceedings Vol. 49 (North-Holland, Amsterdam, 1990)
M. Stone, Nucl. Phys. B314, 557 (1989)
M. V. Berry, Proc. R. Soc. London A392, 45 (1984)
L. Baulieu, Phys. Lett. B 232, 479 (1989)
L. Baulieu, Phys. Lett. B 232, 473 (1989)
Yue Yu, Phys. Rev. D 40, 1301 (1989)
L. Cugliandolo, G. Lozano, and F. A. Schaposnik, Phys. Lett. B 253, 90 (1991)
G. T. Horowitz, Commun. Math. Phys. 125, 417 (1989)
F. Ferrari and H. Hüffel, Phys. Lett. B 261, 47 (1991)
J. Klauder, Phys. Rev. D 19, 2349 (1979) L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)
G. Parisi and Y. S. Wu, Sci. Sinica 24, 484 (1981)
I. A. Batalin and G. A. Vilkovisky, Phys. Rev. D 28, 1 (1983)
K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979) ibid.44, 1733 (1980) Phys. Rev. D 21, 2848 (1980)
N. K. Nielsen and P. van Nieuwenhuizen, Phys. Rev. D 38, 3183 (1988)
L. Cugliandolo, G. Lozano, and F. A. Schaposnik, Phys. Lett. B 244, 249 (1990)
F. A. Schaposnik, in III Meeting on Quantum Mechanics of Fundamental Systems, edited by C. Teitelboim and J. Zanelli (Plenum, New York, 1991)
L. Baulieu and I. M. Singer, in Conformal Field Theories and Related Topics, LAPP, Proceedings of the Conference, Annecy, France, 1988, edited by P. Binétruy, P. Sorba, and R. Stora [Nucl. Phys. (Proc. Suppl.) 5B, 18 (1988)]
J. M. Labastida and M. Pernici, Phys. Lett. B 212, 21 (1988)
A. Belavin and A. M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 22, 503 (1975) [JETP Lett. 22, 245 (1975)]
K. Iwasaki, Prog. Theor. Phys. 81, 1104 (1989)
D. Birmingham, M. Rakowski, and G. Thompson, Phys. Lett. B 212, 187 (1988) ibid.214, 381 (1988)
J. L. Gervais and B. Sakita, Phys. Rev. D 11, 2943 (1975) A. M. Polyakov, Nucl. Phys. B120, 429 (1977)
T. P. Killingback, Commun. Math. Phys. 100, 481 (1985)
A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986)

[c1] E. Witten, Commun. Math. Phys. 117, 353 (1988) ibid.118, 411 (1988) ibid.121, 351 (1989)

[c2] E. Witten, Phys. Lett. B 206, 601 (1988)

[c3] F. D. M. Haldane, Phys. Lett. 93A, 464 (1983)

[c4] P. B. Wiegmann, Phys. Rev. Lett. 60, 821 (1988)

[c5] E. Fradkin and M. Stone, Phys. Rev. B 38, 7215 (1988)

[c6] A. M. Polyakov, in Fields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School, Les Houches, France, 1988, edited by E. Brezin and J. Zinn-Justin, Les Houches Summer School Proceedings Vol. 49 (North-Holland, Amsterdam, 1990)

[c7] M. Stone, Nucl. Phys. B314, 557 (1989)

[c8] M. V. Berry, Proc. R. Soc. London A392, 45 (1984)

[c9] L. Baulieu, Phys. Lett. B 232, 479 (1989)

[c10] L. Baulieu, Phys. Lett. B 232, 473 (1989)

[c11] Yue Yu, Phys. Rev. D 40, 1301 (1989)

[c12] L. Cugliandolo, G. Lozano, and F. A. Schaposnik, Phys. Lett. B 253, 90 (1991)

[c13] G. T. Horowitz, Commun. Math. Phys. 125, 417 (1989)

[c14] F. Ferrari and H. Hüffel, Phys. Lett. B 261, 47 (1991)

[c15] J. Klauder, Phys. Rev. D 19, 2349 (1979) L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)

[c16] G. Parisi and Y. S. Wu, Sci. Sinica 24, 484 (1981)

[c17] I. A. Batalin and G. A. Vilkovisky, Phys. Rev. D 28, 1 (1983)

[c18] K. Fujikawa, Phys. Rev. Lett. 42, 1195 (1979) ibid.44, 1733 (1980) Phys. Rev. D 21, 2848 (1980)

[c19] N. K. Nielsen and P. van Nieuwenhuizen, Phys. Rev. D 38, 3183 (1988)

[c20] L. Cugliandolo, G. Lozano, and F. A. Schaposnik, Phys. Lett. B 244, 249 (1990)

[c21] F. A. Schaposnik, in III Meeting on Quantum Mechanics of Fundamental Systems, edited by C. Teitelboim and J. Zanelli (Plenum, New York, 1991)

[c22] L. Baulieu and I. M. Singer, in Conformal Field Theories and Related Topics, LAPP, Proceedings of the Conference, Annecy, France, 1988, edited by P. Binétruy, P. Sorba, and R. Stora [Nucl. Phys. (Proc. Suppl.) 5B, 18 (1988)]

[c23] J. M. Labastida and M. Pernici, Phys. Lett. B 212, 21 (1988)

[c24] A. Belavin and A. M. Polyakov, Pis'ma Zh. Eksp. Teor. Fiz. 22, 503 (1975) [JETP Lett. 22, 245 (1975)]

[c25] K. Iwasaki, Prog. Theor. Phys. 81, 1104 (1989)

[c26] D. Birmingham, M. Rakowski, and G. Thompson, Phys. Lett. B 212, 187 (1988) ibid.214, 381 (1988)

[c27] J. L. Gervais and B. Sakita, Phys. Rev. D 11, 2943 (1975) A. M. Polyakov, Nucl. Phys. B120, 429 (1977)

[c28] T. P. Killingback, Commun. Math. Phys. 100, 481 (1985)

[c29] A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986)

Physical Review D

Export citation

Export citation

Equivalence of the path-integral theory of spinning particles and the topological nonlinear $σ$ model in $d = 2$ dimensions

Abstract

References (29)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Equivalence of the path-integral theory of spinning particles and the topological nonlinear σ model in d=2 dimensions

Abstract

References (29)

Equivalence of the path-integral theory of spinning particles and the topological nonlinear $σ$ model in $d = 2$ dimensions