ON GEOMETRIC QUANTIZATION FOR THE BOSONIC STRINGS(1)

YU Yue; GUO Han-Ying

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2024年10月30日

Chinese Physics C> 1989, Vol. 13> Issue(5) : 419-428

ON GEOMETRIC QUANTIZATION FOR THE BOSONIC STRINGS(1)

YU Yue ^, ,
GUO Han-Ying

Institute of High Energy Physics,Academia Sinica,Beijing2 Institute of Theoretical Physics,Academia Sinica,Beijing

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Abstract：
The geometric quantization for bosonic strings is discussed in this paper.Relations among different polarizations and representations of operators in different polarizations are given.It is pointed out that the prequantization Hilbert space is the unitary representation of the conformal group where the centre term of Virasoro algebra does not exist but this representation is reducible.By polarization it is reduced into two projective representations with the phase factors with opposite signs.Then the conformal anomaly is obtained.In the viewpoint of geometric quantization,the emergence of the conformal anomaly stems from the fact that polarization is introduced because the quantum states of string must satisfy the uncertainty relation but all generators of conformal transformation don't preserve the same polarization.

References

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J. Scherk, Rev. Mod. Phys., 17(1975), 123;P. Goddard and C. B. Thorn, Phys. Lett., 40B(1972), 235.[2] J. Sniatycki. Geometric Quantization and Quantum Mechanics (Springer-Verlag. New York (1980).[3] N. J. Woodhouse, Geometric Quantization (Clarendon Press, Oxford, UK (1980).[4] M. Bowick and S. G. Rajeev, Nucl. Phys., B293(1987), 348; Phys. Rev. Lett., 58(1987), 353; MIT preprints CTP 1494(1987).D. Harri, D. K. Hong, P. Ramond and V. G. Rodgers, Florida preprints, UFTP-87-10.[5] M. J. Gotay, J. M. Nester and G. Hinds, J. Math. Phys., 19(1978), 2388.M. J. Gotay. J. Math. Phys., 27(1986), 2051.A. Ashtekar and M. Stillerman, J. Math. Phys. 27(1986), 1393.

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YU Yue and GUO Han-Ying. ON GEOMETRIC QUANTIZATION FOR THE BOSONIC STRINGS(1)[J]. Chinese Physics C, 1989, 13(5): 419-428.

YU Yue and GUO Han-Ying. ON GEOMETRIC QUANTIZATION FOR THE BOSONIC STRINGS(1)[J]. Chinese Physics C, 1989, 13(5): 419-428. shu

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Received: 1900-01-01
Revised: 1900-01-01

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沈阳化工大学材料科学与工程学院沈阳 110142

ON GEOMETRIC QUANTIZATION FOR THE BOSONIC STRINGS(1)

YU Yue ^,
GUO Han-Ying

Corresponding author: YU Yue,

Institute of High Energy Physics,Academia Sinica,Beijing2 Institute of Theoretical Physics,Academia Sinica,Beijing

Received Date: 1900-01-01
Accepted Date: 1900-01-01
Available Online: 1989-05-05

Abstract: The geometric quantization for bosonic strings is discussed in this paper.Relations among different polarizations and representations of operators in different polarizations are given.It is pointed out that the prequantization Hilbert space is the unitary representation of the conformal group where the centre term of Virasoro algebra does not exist but this representation is reducible.By polarization it is reduced into two projective representations with the phase factors with opposite signs.Then the conformal anomaly is obtained.In the viewpoint of geometric quantization,the emergence of the conformal anomaly stems from the fact that polarization is introduced because the quantum states of string must satisfy the uncertainty relation but all generators of conformal transformation don't preserve the same polarization.