Prog. Theor. Phys. Vol. 56 No. 4 (1976) pp. 1295-1309

[ Full Text PDF : FREE ACCESS (1088K) ]

Singular Lagrangian and the Dirac-Faddeev Method

— Existence Theorem of Constraints in 'Standard Form' —

Toshihide Maskawa^* and Hideo Nakajima^**

^*Department of Physics, Kyoto University, Kyoto 606
^**Research Institute for Fundamental Physics, Kyoto University, Kyoto 606

(Received April 12, 1976)

Abstract:

Rigorous basis is given to the Dirac-Faddeev method, i.e., the Feynman integral with constraints (1st-class and/or 2nd-class), by proving existence theorem of constraints in 'standard form' for the restricted submanifold of definite dimension. The theorem states that such canonical variables exist for a given restricted submanifold that the submanifold is specified by putting those canonical variables equal to zero. This theorem will also give straightforward understanding of concepts in the Dirac formalism, e.g., Dirac bracket.

URL : http://ptp.ipap.jp/link?PTP/56/1295/
DOI : 10.1143/PTP.56.1295

References:

P. A. M. Dirac, Can. J. Math. 2 (1950), 129; Proc. Roy. Soc. A 246 (1958), 326.
P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yeshiva University, (New York, 1964).
L. D. Faddeev, Teoret. i Mat. Fiz. 1 (1969), 3.
A. J. Hanson, T. Regge and C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale dei Lincei, Rome, 1976).
P. Senjanovic, preprint (CCNY-HEP-75/6).
H. Yabuki, preprint (RIMS-183).
H. Flanders, Differential Forms with Applications to the Physical Sciences (Academic Press Inc., New York, 1963).
R. Abraham, Foundations of Mechanics (W. A. Benjamin Inc., 1967).
P. Senjanovic, preprint (CCNY-HEP-75/6).
P. G. Bergmann and I. Goldberg, Phys. Rev. 98 (1955), 531[APS].
T. Yamanouchi, R. Utiyama and T. Nakano, Theory of General Relativity and Gravitation (Syokabo, 1967).

Citing Article(s) :

Progress of Theoretical Physics Vol. 57 No. 4 (1977) pp. 1416-1425 :
Null-Plane Quantization in Non-Linear Lagrangian Theories

Yoshimasa Nakano
Progress of Theoretical Physics Vol. 63 No. 2 (1980) pp. 599-615 :
Hamiltonian Formalism for Systems Including Grassmann Numbers and Canonical Quantization by Feynman Path-Integral with Constraints

Yoshimasa Nakano
Progress of Theoretical Physics Vol. 72 No. 2 (1984) pp. 350-365 :
Sthochastic Quantization of Constrained Systems

Mikio Namiki, Ichiro Ohba and Keisuke Okano
Progress of Theoretical Physics Vol. 73 No. 1 (1985) pp. 186-196 :
Numerical Simulation of Nonlinear σ-Model by Means of Stochastic Quantization Method

Mikio Namiki, Ichiro Ohba, Keisuke Okano, Masanori Rikihisa and Satoshi Tanaka
Progress of Theoretical Physics Vol. 73 No. 4 (1985) pp. 1025-1042 :
Relation between Generators of Gauge Transformations and Subsidiary Conditions on State Vectors

Reiji Sugano, Tohru Sohkawa and Toshiei Kimura
Progress of Theoretical Physics Vol. 83 No. 3 (1990) pp. 379-384 :
Hermitian Fermi Coordinates, Dimensions and Path Integrals

Hideo Nakajima
Progress of Theoretical Physics Vol. 85 No. 6 (1991) pp. 1347-1361 :
Commutators in Supersymmetric CP^n-1₂ Model

Masayoshi Nakamura and Hiroaki Minowa
Progress of Theoretical Physics Vol. 95 No. 1 (1996) pp. 203-225 :
Bosonization of Thirring Model in Arbitary Dimension

Kenji Ikegami, Kei-ichi Kondo and Atsushi Nakamura
Progress of Theoretical Physics Supplement No.111 (1993) pp. 53-62 :
Stochastic Quantization of Constrained Systems

Ichiro Ohba