Relativistic Quantum Mechanics and Fields
The Dirac equation is a relativistic equation for describing spin-1/2 particles. Consider the Dirac equation for a free particle with mass M. The Schrödinger-like form is an equation of first order in both space and time, given by (we use natural units wh
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L.N. Savushkin H. Toki
The Atomic Nucleus as a Relativistic System With 45 Figures and 35 Tables
,
Springer
Professor Lev N. Savushkin St. Petersburg University for Telecommunications Department of Physics nab.r.Moika, 61 191186 St. Petersburg, Russia E-Mail: [email protected]
Professor Hiroshi Toki Osaka University, RCNP Mihogaoka 10-1, Ibaraki Osaka 567-0047, Japan E-mail: [email protected]
Library of Congress Cataloging-in-Publication Data. Savushkin, L.N. (Lev Nikolaevich), 1939The atomic nucleus as a relativistic system / L.N. Savushkin, H. Toki. p.cm. - (Texts and monographs in physics, ISSN 0172-5998) Includes bibliographical references and index. ISBN 978-3-642-07347-2 ISBN 978-3-662-10309-8 (eBook) DOI 10.1007/978-3-662-10309-8 I. Nuclear physics. 2. Relativity (Physics) I. Toki. H. II. Title. III. Series. QC782.S382003 539.7- 1, one obtains the linear Klein-Gordon equation (2.53). If we use (2.69) for n
> 3, we arrive at the well-known equation (2.70)
of what is called the cp4 theory. It plays an essential role in gauge theories (see below). We have defined above the dynamical field variables '¢c,,(x) and 7ra:(x). Now, in accordance with the second-quantization procedure [3], let us impose the well-known equal-time canonical commutation and anticommutation relations for Bose and Fermi fields, respectively: [7ra:(X), '¢,8(x')]8(t - t') = -i8a:,88(x - x') , ['¢a:(x), '¢,8(x')]8(t - t') = [7ra:(x), 7r,8(x')]8(t - t')
=0
(2.71) (2.72)
2.3 Relativistic Free-Field Theories
15
and {7ra (X), 'I/J,a(X')} 8(t - t') = i8a,a8(x - x') , {'I/Ja(x),'I/J,a(x')} 8(t - t') = {7ra (X) , 7r,a(x')} 8(t - t') = 0 .
(2.73) (2.74)
These equations can be easily generalized to arbitrary spacelike [(x-x')2 < 0] intervals also [3]. The introduction of (2.71)-(2.74) corresponds to replacing the dynamical variables 'l/Ja(x) and 7ra (x) by the respective Hermitian operators. Consider the tensor (2.75) which will be referred to as the canonical energy-momentum tensor. The Euler-Lagrange equations (2.49) ensure the conservation of this quantity: lJ
= 0,1,2,3 .
(2.76)
Let us mention that the conservation law given by (2.76) is related, in accordance with Noether's theorem, to the invariance of the action principle 88 = 0 under transformations of the following type: where
E;I-'
= const,
(2.77)
i.e. under space-time translations. In particular, let us consider
Too
= TOO = 7r a(x)8o'I/Ja(x) -
C(x)
= 1i(x) ,
(2.78)
so that (2.79) determines H, i.e. the Hamiltonian of the field (the total field energy). Let us introduce also the four-momentum pI-'
=
(H,P) =
gl-'V Pv
(2.80)
defined by (2.81) This four-vector Pv appears to be a constant of the motion. Consider the real scalar field as an example. In this case the field is described by a single-component wave function '2 (-q)Pau>.~ (q ')F~l)u>'l (q),
(3.22)
where u(q) is the Dirac spinor for a free nucleon with momentum q, Fa is the vertex op
