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A note on relativistic second quantization

Published online by Cambridge University Press:  24 October 2008

T. S. Chang
T. S. Chang
Affiliation:
Fitzwilliam HouseCambridge
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Extract

The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 2 , April 1947 , pp. 183 - 195
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

† Dirac, P. A. M., Proc. Roy. Soc. A, 180 (1942), 1.CrossRefGoogle Scholar

‡ It is suggested that such particles are called ‘fermions’, which is simpler than ‘Fermi-Dirac particles’ or ‘particles satisfying Fermi-Dirac statistics’.

§ Dirac, P. A. M., Proc. Cambridge Phil. Soc. 35 (1939), 416.CrossRefGoogle Scholar

∥ It seems perhaps hardly necessary to point out the fact that any relativistic theory of second quantization on certain particles cannot proceed unless we know the wave equations which these particles satisfy, in full contrast with a non-relativistic second quantization.

† If we restrict 〈x′|〉 〈x′…|〉 to be of the type

(i.e. restricting ourselves to positive energy particles only), formula (12) remains but with – iΔ(x′–x″) replaced by

† We have not however, extended the 〈ξ′|〉 representations for bosons to fermions. For besons, the definition of 〈ξ′|〉 is

To extend these to fermions, one must assume that the in the above equations anticommute, which spoils the advantages of introducing such representatives.

‡ Fock, V., Z. Phys. 75 (1932), 622.CrossRefGoogle Scholar

§ This discussion can be applied to quantum electrodynamics with ∂A μ/∂x/μ = 0(Chang, T.S., Proc. Roy. Soc. A, 185 (1946), 192)CrossRefGoogle Scholar. By using the above discussion, we may deduce the expressions for etc., given there.