The algebra and the calculus pertaining to a certain class of representations of the de Sitter group is developed. This permits us to formulate covariant field equations in de Sitter space, and, in particular, to construct quantum mechanical equations of motion associated with particles of given spin. The gauge principle is invoked and a spin‐two field emerges, which we identify with the gravitational field. Its coupling to sources is discussed and conservation laws are derived. The emerging nongeometric theory of gravitation is compared with both the Einstein‐Riemann type and other previously proposed nongeometrical theories.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
Some of the ideas developed below were first expressed by us in an essay written for the 1965 Gravity Research Foundation Contest. We are indebted to the Foundation for awarding Honorable Mention to this essay.
9.
10.
and Max Planck Festschrift, p. 339 (Berlin, 1958);
F.
Gürsey
and T. D.
Lee
, Proc. Natl. Acad. Sci. U.S.
49
, 179
(1963
);C.
Fronsdal
, Rev. Mod. Phys.
37
, 221
(1965
); and the literature quoted in these papers.11.
Lower case latin indices from the beginning of the alphabet run from 1 to 5.
12.
Greek indices from the end of the alphabet run from 1 to 4.
13.
Square brackets around indices denote antisymmetrization.
14.
E.
İnönü
and E. P.
Wigner
, Proc. Natl. Acad. Sci. U.S.
39
, 510
(1953
);E.
İnönü
and E. P.
Wigner
, 40
, 119
(1954
).15.
The correct definition of raising and lowering general indices (and in particular the case of doing that with the indices belonging to the adjoint representation) are given in the Appendix.
16.
For tensorial representations, and are integers and can be thought of as the number of nodes in the first and second row of the Young diagram pertaining to the symmetric group
17.
In general, the Lorentz subgroup has the representations labeled by contained in Thus, in the present case,
18.
More about the reduction is said at the beginning of Sec. 3.
19.
In general, upper case Latin indices from the beginning of the alphabet run from 0 to 10, and are used to identify components of the adjoint representation
20.
In general, Latin lower case indices from the end of the alphabet denote components of an arbitrary representation, and run from 1 to d.
21.
That is, keeping only the leading term in R.
22.
Units:
23.
24.
The only exception is the trivial one‐dimensional representation. However, no physical particle corresponds to this, because in view of (32), the field associated with this case is a constant.
25.
See Ref. 7. Actually, we derive the gravitational field from a gauge principle.
26.
Note that in consequence of (35), .
27.
28.
Although (apart from some points of presentation) the preceding paragraphs are standard knowledge, we found it worthwhile to put them in context. More details can be found, for example, in an article by B. S. DeWitt, in Relativity, Groups, and Topology (Gordon and Breach Science Publishers, Inc., New York, 1964), p. 585.
29.
We can say that is the antisymmetric part of whereas is its symmetric part. Note that in a non‐Cartesian system tensors of do not have definite symmetry.
30.
Since we take the gravitational field to be minimal mass field, Eq. (33) gives for the mass of the graviton
31.
From the manipulations explained in the Appendix, it follows that
32.
Going back, say, to Eqs. (58) and (52).
33.
(rs) means symmetrization.
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© 1966 The American Institute of Physics.
1966
The American Institute of Physics
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