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On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit

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On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit P H YS I CA L R EV I E%' VOL UM E 78, N UM 8ER A P R I L 1, 1990 On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit LEsLIE L. FoLDY Case Institute of Technology, Cleveland, Ohio SIEGERIED A. WQUTHUYsENf Universe'ty of Rochester, Rochester, New York (Received November 25, 1949) By a canonical transformation on the Dirac Hamiltonian for a conventional ones which pass over into the position and spin free particle, a representation of the Dirac theory is obtained in operators in the Pauli theory in the non-relativistic limit. which positive and negative energy states are separately repre- The transformation of the new representation is also made in sented by two-component wave functions. Playing an important the case of interaction of the particle with an external electro- role in the new representation are new operators for position and magnetic field. In this way the proper non-relativistic Hamiltonian the Pauli-Hamiltonian) is obtained in the non- spin of the particle which are physically distinct from these (essentially relativistic limit. The same methods be to a Dirac operators in the conventional representation. The components of may applied particle interacting with any type of external field (various meson the time derivative of the new position operator all commute and fields, for example) and this allows one to find the proper non- have for eigenvalues all values between —c and c. The new spin relativistic Hamiltonian in each such case. Some light is cast on operator is a constant of the motion unlike the in the spin operator the question of why a Dirac electron shows some properties conventional representation. By a comparison of the new Hamil- characteristic of a particle of finite extension by an examination tonian with the non-relativistic Pauli-Hamiltonian for particles of of the relationship between the new and the conventional position spin ~, one finds that it is these new operators rather than the operators. INTRODUCTION eigenfunctions, the first two (or upper) components ~ vanish with vanishing momentum. FREE Dirac particle of mass rn is described by a 4' four-component wave function 4 satisfying the Of course any spinor may be split into the sum of ~ ~ com- Dirac equation, two spinors 4 and X, having only upper and lower ponents respectively, in the following manner: H+= (Pm+a p)% =i(8+/Bt), (1) @=I+X, C = (1+P/2)4', X= (1—P/2)%', where is the momentum operator for the particle, p but the spinors 4 and X do not represent states of e and are the well known Dirac matrices (assumed P definite energy in the above representation. here to be in their usual representation with P diagonal), In the non-relativistic limit, where the momentum of and units have been used in which h and c are unity. the particle is small compared to m, it is well known The eigenfunctions of the Hamiltonian operator satisfy that a Dirac particle (that is, one with spin —.', ) can be the equation: described by a two-component wave function in the (8m+~ p)4=~4. Pauli theory. The usual method of demonstrating that the Dirac theory goes into the Pauli theory in this ~'*. The eigenfunctions are of the form u(p)e For each limit makes use of the fact noted above that two of value of the momentum p there are four linearly inde- the four Dirac-function components become small when pendent spinors' u(p) corresponding to the two eigen- the momentum is small. One then writes out the values &(m'+ p') & of the energy and the two eigenvalues equations satisded by the four components and solves, &1 of the s component of an operator X (dehned later approximately, two of the equations for the small corn- in Eq. (26)) related to the spin operator for the particle. ponents. By substituting these solutions in the remain- Except in certain trivial cases, for a given sign of the ing two equations, one obtains a pair of equations for energy at least three of the components of N(p) are the large components which are essentially the Pauli difI'erent from zero. However, two of the four com- spin equations. ' ponents go to zero as the momentum goes to zero, while The above method of demonstrating the equivalence at least one of the other two components remains finite. of the Dirac and Pauli theories encounters difhculties, In the above representation, for positive-energy eigen- however, when one wishes to go beyond the lowest order functions, the last two (or lower) components vanish approximation. One then 6nds that the "Hamiltonian" with vanishing momentum, while for negative-energy associated with the large components is no longer Her- mitian in the presence of external fields because of the * Supported by the AEC and the ONR. appearance of an "imaginary electric moment" for the t Now at the Zeeman Laboratory, University of Amsterdam, Amsterdam, Holland. particle. Furthermore, all four components must again ' See, for example, %.Heitler, The Quantum Theory of Radiation be used in calculating expectation values of operators to (Oxford University Press, New York, 1944), Chapter 3. Ke shall use the term spinor for any four-component column (or row) ' See, f'or example, W. Pauli, Handbuch der I'hysik, 2. AuQ. , Bd. matrix occurring in the Dirac theory. 24, Teil 1. 29 L. L. FOLDY AND S. A. KOUTHUYSEN order sz/c'. Apart from these difEculties, however, there teristic of a particle with 6nite extension of the order are some serious questions concerning the properties of of its Compton wave-length. corresponding operators in the Dirac theory and the (5) In the presence of interaction, such as with an Pauli theory even in lowest order. Thus, in the Dirac external electromagnetic Geld, one can still make a theory the operator representing the velocity of the transformation which leads to a representation involv- particle is the operator e whose components have only ing two-component wave functions. The transforma- the eigenvalues ~1. On the other hand, in the Pauli tion, which previously could be made exactly, must now theory the operator representing the velocity of the be made by an in6nite sequence of transformations particle is taken to be p/m whose components have which process leads to a Hamiltonian which is an eigenvalues embracing all real numbers. Furthermore, in6nite series in powers of (1/m), where m is the mass of different components of the velocity operator in the the particle. This series is presumably semi-convergent Dirac theory do not commute and therefore are not in the sense that for given external 6elds, a 6nite simultaneously measurable with arbitrary precision, number of terms of the series is a better-and-better while different components of the velocity operator in approximation to the exact Hamiltonian, the larger the the Pauli theory do commute and can therefore be value of m. For strong interactions which strongly measured simultaneously with arbitrary precision. One couple free-particle states of positive and negative can well ask how the operators which purportedly repre- energy, this representation is of little value; however, sent the same physical variable in the two theories for sufticiently weak interactions, a 6nite number of can have such completely different properties. terms of the series may be employed to obtain relativistic One can conclude from the above discussion that the corrections to any order in (1/m). In this way one can relation between the Dirac theory and the Pauli theory obtain the proper non-relativistic limit for the Hamil- is by no means clear from the usual method of de- tonian representing a Dirac particle interacting with scending from four- to two-component wave functions, any type of external Geld. and that further clarjL6cation of the connection between the theories would be desirable. In what follows we THE FREE DIRAC PARTICLE present an alternative method for passing from four- to The essential reason why four components are in two-component wave functions in the Dirac theory general necessary to describe a state of positive or which alternative method clarifies many of the questions negative energy in the representation of the Dirac left open by the usual treatment. The method consists theory corresponding to Eq. (1) is that the Hamiltonian of a transformation to a new representation for the in this equation contains odd operators, ' Dirac theory, putting the theory in a form closely specifically the components of the operator e. If it is possible to analogous to the Pauli theory and permitting a direct perform a canonical transformation on Eq. (1) which comparison of the two. Specifically, we shall show that: brings it into a form which is free of odd operators, then (1) For a free Dirac particle there exists a representa- it will be possible to represent positive- and negative- tion of the Dirac theory in which, for both relativistic energy states by wave functions having only two com- and non-relativistic energies, positive-energy states and ponents in each case, the other pair of components negative-energy states are separately described by two- being identically zero. We shall now show that there component wave functions. exists a canonical transformation which accomplishes (2) There exists in the Dirac theory another position this end. operator than the usual one; this operator has the just Hermitian transforma- property that its time derivative is the operator If S is an operator, then the & tion, p/(m'+P') for positive-energy states and —y/(m'+ p'-) & for negative-energy states corresponding to the con- 0"—e.s@ (3) ventional concept of the velocity of the particle.
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