Reformulation of the Dirac-Fierz-Pauli Equation for Spin 3/2

S. Tani

doi:10.1103/PhysRevD.2.980

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Reformulation of the Dirac-Fierz-Pauli Equation for Spin 3/2

S. Tani

Physics Department, Marquette University, Milwaukee, Wisconsin 53233

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Phys. Rev. D 2, 980 – Published 15 September, 1970

DOI: https://doi.org/10.1103/PhysRevD.2.980

Abstract

We study the $(1, \frac{1}{2}) \oplus (\frac{1}{2}, 1)$ irreducible representation of the homogeneous Lorentz group thoroughly. Though our starting point is the original version of the spinor equation due to Dirac and Fierz, we resort to an algebraic method and derive all necessary transformation functions and projection operators. Consequently, we can see a physical significance of each mathematical operation immediately. There are two kinds of particles in this representation: One has mass $m$ and spin $\frac{3}{2}$ in the rest system, the other mass $2 m$ and spin ½. A projection operator to a subspace for each type of particle is Lorentz invariant. There are four $γ$ matrices which transform like components of a four-vector. The inverse of the operator which defines the equation of motion for a free particle will define the (Lorentz-invariant) propagator. Unfortunately, it is hard to carry out the canonical quantization, and we cannot derive the same propagator by a canonical method. The algebraic method suggested here will be very useful when we extend the theory by adding a $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ representation, as is done by Pauli and Fierz, Gupta, and others.

References (33)

P. A. M. Dirac, Proc. Roy. Soc. (London) A155, 447 (1936)
M. Fierz, Helv. Phys. Acta 12, 3 (1939)
A. Z. Capri, Phys. Rev. 178, 2427 (1969)
H. Joos, Fortschr. Physik 10, 65 (1962)
S. Weinberg, Phys. Rev. 133, B1318 (1964) ibid.134, B882 (1964) ibid.181, 1893 (1969)
D. L. Weaver, C. L. Hammer, and R. H. Good, Jr., Phys. Rev. 135, B241 (1964)
D. Shay, H. S. Song, and R. H. Good, Jr., Nuovo Cimento Suppl. 3, 455 (1965)
M. Fierz and W. Pauli, Proc. Roy. Soc. (London) A173, 211 (1939)
S. N. Gupta, Phys. Rev. 95, 1334 (1954)
A. S. Wightman, in Symmetry Principles at High Energies, edited by A. Perlmutter, C. A. Hurst, and B. Kurşunoǧlu (Benjamin, New York, 1968), p. 291
W. Rarita and J. Schwinger, Phys. Rev. 60, 611 (1941)
H. A. Kramers, F. J. Belinfante, and J. K. Lubanski, Physica 8, 597 (1941) J. K. Lubanski, ibid. 9, 310 (1942)
V. Bargamann and E. P. Wigner, Proc. Natl. Acad. Sci. U. S. 34, 211 (1948)
H. J. Bhabha, Rev. Mod. Phys. 17, 200 (1945) ibid.21, 451 (1949)
Harish-Chandra, Phys. Rev. 71, 793 (1947) Proc. Roy. Soc. (London) A192, 195 (1947)
E. M. Corson, Introduction to Tensors, Spinsors, and Relativistic Wave-Equations (Hafner, New York, 1953)
H. Umezawa, Quantum Field Theory (North-Holland, Amsterdam, 1956)
L. L. Foldy, Phys. Rev. 102, 568 (1956)
A. D. Bryden, Nucl. Phys. 53, 165 (1964) ibid.58, 314 (1964)
D. L. Pursey, Nucl. Phys. 53, 174 (1964)
D. L. Pursey, Ann. Phys. (N. Y.) 32, 157 (1965)
T. J. Nelson and R. H. Good, Jr., Rev. Mod. Phys. 40, 508 (1968)
P. M. Mathews, Phys. Rev. 155, 1415 (1967)
A. Pais, Rev. Mod. Phys. 38, 215 (1966) A. Salam, R. Delbourgo, and J. Strathdee, Proc. Roy. Soc. (London) A284, 146 (1965)
W. K. Tung, Phys. Rev. 156, 1385 (1967)
M. D. Scadron and H. F. Jones, Phys. Rev. 173, 1734 (1968)
S. J. Chang, Phys. Rev. 161, 1316 (1967)
[16] M. Hamermesh, Group Theory (Addison-Wesley, Reading, Mass., 1962), p. 353 B. L. van der Waerden, Die Gruppentheoretische Methode in der Quantenmechanik (Springer, Berlin, 1932), Secs. 17 and 20
L. I. Schiff, 3rd ed., Quantum Mechanics (McGraw-Hill, New York, 1968), p. 203
[21], pp. 167 and 168
A. Aurilia and H. Umezawa, Phys. Rev. 182, 1682 (1969)
S. Tani, Ann. Phys. (N. Y.) 37, 411, 451 (1966) Phys. Rev. 174, 2054 (1968)
H. Jehle and W. C. Parke, Phys. Rev. 137, B760 (1965)

[c1] P. A. M. Dirac, Proc. Roy. Soc. (London) A155, 447 (1936)

[c2] M. Fierz, Helv. Phys. Acta 12, 3 (1939)

[c3] A. Z. Capri, Phys. Rev. 178, 2427 (1969)

[c4] H. Joos, Fortschr. Physik 10, 65 (1962)

[c5] S. Weinberg, Phys. Rev. 133, B1318 (1964) ibid.134, B882 (1964) ibid.181, 1893 (1969)

[c6] D. L. Weaver, C. L. Hammer, and R. H. Good, Jr., Phys. Rev. 135, B241 (1964)

[c7] D. Shay, H. S. Song, and R. H. Good, Jr., Nuovo Cimento Suppl. 3, 455 (1965)

[c8] M. Fierz and W. Pauli, Proc. Roy. Soc. (London) A173, 211 (1939)

[c9] S. N. Gupta, Phys. Rev. 95, 1334 (1954)

[c10] A. S. Wightman, in Symmetry Principles at High Energies, edited by A. Perlmutter, C. A. Hurst, and B. Kurşunoǧlu (Benjamin, New York, 1968), p. 291

[c11] W. Rarita and J. Schwinger, Phys. Rev. 60, 611 (1941)

[c12] H. A. Kramers, F. J. Belinfante, and J. K. Lubanski, Physica 8, 597 (1941) J. K. Lubanski, ibid. 9, 310 (1942)

[c13] V. Bargamann and E. P. Wigner, Proc. Natl. Acad. Sci. U. S. 34, 211 (1948)

[c14] H. J. Bhabha, Rev. Mod. Phys. 17, 200 (1945) ibid.21, 451 (1949)

[c15] Harish-Chandra, Phys. Rev. 71, 793 (1947) Proc. Roy. Soc. (London) A192, 195 (1947)

[c16] E. M. Corson, Introduction to Tensors, Spinsors, and Relativistic Wave-Equations (Hafner, New York, 1953)

[c17] H. Umezawa, Quantum Field Theory (North-Holland, Amsterdam, 1956)

[c18] L. L. Foldy, Phys. Rev. 102, 568 (1956)

[c19] A. D. Bryden, Nucl. Phys. 53, 165 (1964) ibid.58, 314 (1964)

[c20] D. L. Pursey, Nucl. Phys. 53, 174 (1964)

[c21] D. L. Pursey, Ann. Phys. (N. Y.) 32, 157 (1965)

[c22] T. J. Nelson and R. H. Good, Jr., Rev. Mod. Phys. 40, 508 (1968)

[c23] P. M. Mathews, Phys. Rev. 155, 1415 (1967)

[c24] A. Pais, Rev. Mod. Phys. 38, 215 (1966) A. Salam, R. Delbourgo, and J. Strathdee, Proc. Roy. Soc. (London) A284, 146 (1965)

[c25] W. K. Tung, Phys. Rev. 156, 1385 (1967)

[c26] M. D. Scadron and H. F. Jones, Phys. Rev. 173, 1734 (1968)

[c27] S. J. Chang, Phys. Rev. 161, 1316 (1967)

[c28] [16] M. Hamermesh, Group Theory (Addison-Wesley, Reading, Mass., 1962), p. 353 B. L. van der Waerden, Die Gruppentheoretische Methode in der Quantenmechanik (Springer, Berlin, 1932), Secs. 17 and 20

[c29] L. I. Schiff, 3rd ed., Quantum Mechanics (McGraw-Hill, New York, 1968), p. 203

[c30] [21], pp. 167 and 168

[c31] A. Aurilia and H. Umezawa, Phys. Rev. 182, 1682 (1969)

[c32] S. Tani, Ann. Phys. (N. Y.) 37, 411, 451 (1966) Phys. Rev. 174, 2054 (1968)

[c33] H. Jehle and W. C. Parke, Phys. Rev. 137, B760 (1965)

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Reformulation of the Dirac-Fierz-Pauli Equation for Spin 3/2

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