Abstract
Schwinger's action principle is formulated for the quantum system which corresponds to the classical system described by the LagrangianL c(\(\dot x\), x)=(M/2)gij(x)\(\dot x\) i \(\dot x\) j−v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider onlyc-number variations of coordinates and time. The Euler-Lagrange equation, the canonical commutation relations, and the canonical equations of motion are derived from this principle in a consistent manner. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant. The judicious choice of the quantal Lagrangian is essential in our formulation. A quantum mechanical analog of Noether's theorem, which relates the invariance of the quantal action with a conservation law, is established. The ambiguities in the quantal Lagrangian are also discussed and it is pointed out that the requirement of invariance is not sufficient to determine uniquely the quantal Lagrangian and the Hamiltonian.
Similar content being viewed by others
References
T. Kawai,Prog. Theoret. Phys. 48, 2082 (1972).
T. Kawai,Prog. Theoret. Phys. 49, 1777 (1973).
H. Kamo and T. Kawai,Prog. Theoret. Phys. 50, 680 (1973).
H. Kamo and T. Kawai,Lett. Nuovo Cimento 7, 571 (1973).
B. S. DeWitt,Rev. Mod. Phys. 29, 377 (1957).
W. Yourgrau and S. Mandelstam,Variational Principles in Dynamics and Quantum Theory, 3rd ed. (W. B. Saunders, Philadelphia and Pitman, London, 1968).
R. P. Feynman,Rev. Mod. Phys. 20, 367 (1948).
I. W. Mayes and J. S. Dowker,J. Math. Phys. 14, 434 (1973).
K. S. Cheng,J. Math. Phys. 13, 1723 (1972).
K. S. Cheng,J. Math. Phys. 14, 980 (1973).
L. Cohen,J. Math. Phys. 11, 3296 (1970).
F. J. Testa,J. Math. Phys. 12, 1471 (1971).
G. S. Um,J. Math. Phys. 15, 220 (1974).
J. Schwinger,Phys. Rev. 82, 914 (1951).
J. M. Charap,J. Phys. 6, 393 (1973).
T. Kimura, T. Ohtani, and R. Sugano,Prog. Theoret. Phys. 48, 1395 (1972).
F. J. Bloore, L. Routh, and J. Underhill,Nucl. Phys. B 55, 637 (1973).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kawai, T. Quantum action principle in curved space. Found Phys 5, 143–158 (1975). https://doi.org/10.1007/BF01100323
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01100323