On the theory of the magnetic electron

by Lev Landau

On the theory of the magnetic electron

Von D. Iwanenko and L. Landau in Leningrad

Received on 8 March 1927

 

Instead of the usual Schrödinger wave function, a series of antisymmetric tensors of different ranks $\psi _{ikl...}$ are introduced. In this way it is possible to set up relativistically invariant equations without any assumption about rotation, which result in the required effects by themselves. Finally, it is shown how a generalization can be obtained for any number of degrees of freedom.

 

1. Introduction. As is well known, many experimental facts, mainly from the field of spectroscopy, necessarily lead to the result that the electron has an additional magnetic moment. Goudsmit and Uhlenbeck tried to explain this peculiar phenomenon by modifying the usual electron model; they believed that we are dealing with a rotation of the electron. Since the corresponding quantum number can assume two values, it is evident that this rotation must be quantized half-count. However, this is in complete contradiction to wave mechanics, which necessarily results in an integer quantization including the rest state for all rotating bodies. The further development of the quantum theory of electron rotation also led to contradictions. In order to do arrive at the correct value of the term splitting, the appropriate relativistic magnetic force expression ${\boldsymbol {H}}+{\frac {1}{c}}{\boldsymbol {E}}\times {\boldsymbol {v}}$ need not be used, but ${\boldsymbol {H}}+{\frac {1}{2c}}{\boldsymbol {E}}\times {\boldsymbol {v}}$ which seems incomprehensible from the point of view of the concept of rotation. The attempts by Frenkel^[1] and Thomas^[2] derive this half on the basis of time averaging cannot be regarded as sufficient, since the half does not appear in the energy expression. Although Darwin^[3] succeeded in setting up the correct equations by introducing not only a scalar but also a $\psi$-vector although the coefficients remained unfounded for him and therefore relativistic invariance was not achieved. The same applies, of course, to Jordan's quaternion method.^[4]

It seems, therefore, that the magnetic form of the electron has nothing to do with rotation, but represents a phenomenon which has it origin much more deeply in the nature of things. We hope to have shown that this "R-effect" can really be derived from very general wave-mechanical ideas and is in no way a special property of the electron. The behaviour of the magnetic moment of the electron to the mechanical moment has no electrodynamic but a quantum character.

Finally, we would like to mention that Dirac,^[5] in a recently published work, also treats the problem from the same starting point. Both theories seem to be equivalent, apart from the complete difference in methods and equations. However, their precise connection is unclear to us. Dirac's theory, in contrast to ours, is hardly applicable to multi-electron problems.

2. We now would like to set up the invariant wave equations for the R-effect. The question arises as to what quantities to choose as wave functions. As can be seen, a single scalar is certainly not enough, since then we get no splitting of terms. Therefore other tensor quantities have to be introduced. It seems natural (the example of the electromagnetic field also reminds us of this) to use only antisymmetric tensors (The antisymmetry should refer to each pair of indices). But t would not be appropriate to limit ourselves to a specific order; as further considerations show, all possible anti-symmetric tensors must be constructed. For our four dimensions, the highest possible order of an antisymmetric tensor is obviously four.

In order to set up the desired equations, we resort to the methods of ordinary wave mechanics. From these we choose the method of the Lagrangian function, because this is the easiest way to calculate the basic quantities of the theory, such as the "current vector", "energy tensor" and others. The fundamental role is played by the velocity vector $u_{k}={\frac {1}{mc}}\left(p_{k}-{\frac {e}{c}}\varphi _{k}\right)$ corresponding to the operator $u_{k}={\frac {1}{mc}}\left({\frac {\hbar }{c}}\nabla _{k}-{\frac {e}{c}}\varphi _{k}\right)$^[6] ($\varphi _{k}$ is the four potential of the external electromagnetic field). As is known, this suffices to set up the equations without using other operators.

If we do not think of $u_{k}$ as an operator but as a simple vector, the Lagrangian function can be constructed as a simple function of $\psi _{ikl...}$, ${\overline {\psi }}_{ikl...}$^[7] and $u_{k}$. In order to get linear equations under variations, we have to start from a bilinear (in $\psi$ and ${\overline {\psi }}$) expression. We would like form only the following scalars from $\psi$, ${\overline {\psi }}$ and $u_{k}$:^[8]

$\psi _{abc...}^{(N)}{\overline {\psi }}_{abc...}^{(N)},$

$u_{a}\psi _{abc...}^{(N+1)}{\overline {\psi }}_{bc...}^{(N)},$

and the complex conjugate:

$u_{a}{\overline {\psi }}_{abc...}^{(N+1)}\psi _{bc...}^{(N)},$

[$(N)$ denotes the order of the tensor $\psi$. The summation is extended only to essentially different terms; i.e., the combinations which arise due to swapping of the indices are not included in the calculation. For example, $\psi _{ab}^{(2)}{\overline {\psi }}_{ab}^{(2)}={\frac {1}{2}}\sum _{a}\sum _{b}\psi _{ab}^{(2)}{\overline {\psi }}_{ab}^{(2)}$.] The requirement that we obtain in the classical limit the relation

$u^{2}+1=0,$

if we want to stick to reality, necessarily leads us to the following values of the coefficients:

$L=\sum _{N}(-1)^{N}\left\{\psi _{abc...}^{(N)}{\overline {\psi }}_{abc...}^{(N)}+u_{a}\psi _{abc...}^{(N+1)}{\overline {\psi }}_{bc...}^{(N)}+u_{a}{\overline {\psi }}_{abc...}^{(N+1)}\psi _{bc...}^{(N)}\right\}.$

In the operator form, we have

(1)$L=\sum _{N}(-1)^{N}\left\{\psi _{abc...}^{(N)}{\overline {\psi }}_{abc...}^{(N)}+{\overline {\psi }}_{bc...}^{(N)}\left[u_{a}\psi _{abc...}^{(N+1)}\right]+\psi _{bc...}^{(N)}\left[u_{a}{\overline {\psi }}_{abc...}^{(N+1)}\right]\right\}.$

First, we would like to use this function to obtain the wave equations. The variation of ${\overline {\psi }}^{(N)}$ gives us

(2)$\psi _{ik...}^{(N)}+u_{a}\psi _{aik}^{(N+1)}-(u_{\alpha }\psi _{\beta \gamma ...}^{(N-1)})_{ik..}=0.$

(The Greek indices $\alpha \beta \gamma ...$ indicate that an antisymmetric tensor is formed from the corresponding expression. For example, $(u_{\alpha }\psi _{\beta \gamma })_{ikl}=u_{i}\psi _{kl}+u_{k}\psi _{li}+u_{l}\psi _{ik}$, $(u_{\alpha }\psi _{\beta })_{ik}=u_{i}\psi _{k}-u_{k}\psi _{i}$). Equations (2) are intended to replace the ordinary Schrödinger equation. As we will show later, they do indeed contain the solution of R-Problems.

Since the number of equations is obviously too large, the question of choosing the solutions arises. For this purpose they have to be completed by additional conditions. Linear constraints are excluded, since the constraints rather have a quadratic character. Its derivation presents considerable difficulties and we have not succeeded so far.

Let us calculate some useful quantities. The variation of the Lagrangian function with respect to the four potential yields for the current vector with suitable normalization

(3)$j_{i}=\sum _{N}(-1)^{N}\left\{{\overline {\psi }}_{ab..}^{(N)}\psi _{iab...}^{(N+1)}+\psi _{ab...}^{(N)}{\overline {\psi }}_{iab...}^{(N+1)}\right\},$

which is applicable in our case. In an analogous way, the variation with respect to the spacetime metric $g_{ik}$ leads to the "energy-momentum tensor".

The H-equations (2) can still be represented in another form. For this, we eliminate $\psi ^{(N+1)}$ and $\psi ^{(N-1)}$ from the $(N-1)$th, $(N)$th and $(N+1)$th equations:

$\psi _{ik...}^{(N)}+u_{a}\left\{-u_{b}\psi _{baik...}^{(N+2)}+(u_{\alpha }\psi _{\beta \gamma ...}^{(N)})_{aik...}\right\}+\left\{u_{\alpha }\left[u_{a}\psi _{\alpha \beta \gamma ...}^{(N)}-(u_{\lambda }\psi _{\mu \nu ...}^{(N-2)})_{\beta \gamma ...}\right]\right\}_{ik...}=0$

which by considering the identities

(a)$(u_{\alpha }\psi _{\beta \gamma ...})_{aik...}=u_{a}\psi _{ik...}-(u_{\alpha }\psi _{a\beta \gamma ...})_{ik...},$

(b)$\left\{u_{\alpha }(u_{\lambda }\psi _{\mu \nu ...})_{\beta \gamma ...}\right\}_{ik...}=(u_{\alpha }u_{\beta }\psi _{\gamma \delta ...})_{ik...},$

and the antisymmetric nature of $\psi$, becomes

$(u^{2}+1)\psi _{ik...}^{(N)}+(u_{a}u_{b}-u_{b}u_{a})\psi _{abik...}^{(N+2)}+\left\{(u_{\alpha }u_{a}-u_{a}u_{\alpha })\psi _{\alpha \beta \lambda ...}^{(N)}\right\}_{ik...}-\left\{(u_{\alpha }u_{\beta }-u_{\beta }u_{\alpha })\psi _{\gamma \delta ...}^{(N-2)}\right\}_{ik...}=0.$^[9]

Since

$u_{a}u_{b}-u_{b}u_{a}={\frac {\hbar }{i}}{\frac {e}{m^{2}c^{3}}}F_{ba}$

where $F$ denotes the electromagnetic field, we finally obtain

(4)$D\psi _{ik...}^{(N)}-{\frac {\hbar e}{2imc}}\left\{F_{ab}\psi _{abik...}^{(N+2)}-(F_{a\alpha }\psi _{\alpha \beta \gamma }^{(N)})_{ik...}-(F_{\alpha \beta }\psi _{\gamma \delta ...}^{(N-2)})_{ik...}\right\}=0.$

Here $D={\frac {1}{2m}}\left({\frac {\hbar }{i}}\nabla _{a}-{\frac {e}{c}}\varphi _{a}\right)^{2}+{\frac {mc^{2}}{2}}$ is the usual Hamilton-Schrödinger operator.

Equations (2), (3) and (4) solve the R-effect for the case of the relativistic one-body problem. We see that no special properties of the electron have been taken into account, so that our reasoning applies to all analogous problems where the external forces have a potential. To speak of a rotation here is completely meaningless, since the equations are strictly valid for a single point. The notion of a rotating point would be strange!

We can generalize the problem with the help of the usual four-dimensional space. As in classical wave mechanics, the idea of a coordinate space with any number of dimensions can be introduced. This should brings us to the solution of the R-effect for the many-body problem. The invariant expansion of our equations permits their application in the case of any coordinate numbers with the only condition that the Hamilton-Schrödinger expression must be quadratic in the moments. But since this last condition is unfortunately not satisfied in the relativistic field problem, we have to choose a detour in order to create the desired generalization. That is, we have to split the equations by considering space and time independently. Each tensor $\psi ^{(N)}$ is divided into two parts, namely, the space part $X^{(N)}$ of the same order and the time part $\Theta ^{(N+1)}$ with an order reduced by one. Thus, we get

(5a)$DX_{ik...}^{(N)}-{\frac {e\hbar }{2imc}}\left\{H_{ab}X_{abik...}^{(N+2)}-(H_{a\alpha }X_{\alpha \beta \gamma }^{(N)})_{ik...}-(H_{\alpha \beta }X_{\gamma \delta ...}^{(N-2)})_{ik...}-E_{a}\Theta _{aik...}^{(N+1)}+(E_{\alpha }\Theta _{\beta \gamma ...}^{(N-1)})_{ik...}\right\},$

(5b)$D\Theta _{ik...}^{(N)}-{\frac {e\hbar }{2imc}}\left\{H_{ab}\Theta _{abik...}^{(N+2)}-(H_{a\alpha }\Theta _{\alpha \beta \gamma }^{(N)})_{ik...}-(H_{\alpha \beta }\Theta _{\gamma \delta ...}^{(N-2)})_{ik...}+E_{a}X_{aik...}^{(N+1)}-(E_{\alpha }X_{\beta \gamma ...}^{(N-1)})_{ik...}\right\}$

where $H_{ik}$ represents the magnetic field tensor and $E_{i}$ the electric field.

If we are only interested in the phenomena originating from the R-effect and not in the relativistic corrections, then the fundamental question arises as to the order of magnitude of various quantities. A closer examination of the basic equations (2) shows that $X$ and $\Theta$ are of the same order of magnitude; as far as the relative order of magnitude of $X$ and $\Theta$ among themselves is concerned, the tensors of even order $(N=0,2,4,...)$ differs from that of odd order $(N=1,3,5,...)$ by a factor ${\frac {v}{c}}$.^[10] A deeper examination suggests that the even orders are decisive.

Now we would like to apply the method of successive approximation. As the zeroth approximation we choose the classical wave equation

(6)$DX_{ik...}^{(2N)}=\left\{{\frac {\hbar }{i}}{\frac {\partial }{\partial t}}-{\frac {\hbar ^{2}}{2m}}\Delta +e\varphi \right\}X_{ik...}^{(2N)}=0.$^[11]

For $\Theta ^{(2N-1)}$ with the same relative accuracy, we have

(7)$D\Theta _{ik...}^{(2N-1)}={\frac {e\hbar }{2imc}}\left\{E_{a}X_{aik...}^{(2N)}-(E_{\alpha }X_{\beta \gamma ...}^{(2N-2)})_{ik...}\right\}$

and a similar equation for $\Theta ^{(2N+1)}$. If we differentiate (6), we get

$D\nabla _{a}X_{aik...}^{(2N)}=eE_{a}X_{aik...}^{(2N)},$

$D(\nabla _{a}X_{\beta \gamma ...}^{(2N)})_{ik...}=e(E_{\alpha }X_{\beta \gamma ...}^{(2N)})_{ik...},$

after neglecting the higher order terms

$E_{a}=-\nabla _{a}\varphi .$

Thus, it follows

(8)$\Theta _{ik...}^{(2N-1)}={\frac {\hbar }{2imc}}\left\{\nabla _{a}X_{aik...}^{(2N)}-(\nabla _{\alpha }X_{\beta \gamma ...}^{(2N-2)})_{ik...}\right\}.$

Now we can take the approximation one degree further. Substituting the expression (8) for $\Theta$ in (6) yields

(9a)$DX_{ik...}^{(2N)}={\frac {e\hbar }{2imc}}\left\{{\mathcal {H}}_{ab}X_{abik...}^{(2N+2)}-({\mathcal {H}}_{a\alpha }X_{a\beta \gamma ...}^{(2N)})_{ik...}+{\frac {\hbar }{2imc}}E_{a}\nabla _{a}X_{ik...}^{(2N)}-({\mathcal {H}}_{\alpha \beta }X_{\gamma \delta ...}^{2N-2)})_{ik...}\right\},$

where

(9b)${\mathcal {H}}_{ik}=H_{ik}+{\frac {\hbar }{2im,c}}(E_{i}\nabla _{k}-E_{k}\nabla _{i})$.

For a single point, we are dealing only with a scalar $X^{(0)}$ and a vector ${\boldsymbol {X}}^{(2)}$ (in three dimensions, as is well known, a second-order antisymmetric tensor is equivalent to a vector). Equations (9a) and (9b) then becomes

(10a)$DX^{(0)}={\frac {e\hbar }{2imc}}\left\{{\mathcal {H}}_{ab}X_{ab}^{(2)}+{\frac {1}{2c}}E_{a}v_{a}X^{(0)}\right\}={\frac {e\hbar }{2imc}}\left\{{\boldsymbol {\mathcal {H}}}:{\boldsymbol {X}}^{(2)}+{\frac {1}{2c}}{\boldsymbol {E}}\cdot {\boldsymbol {v}}X^{(0)}\right\},$

(10b)$D{\boldsymbol {X}}^{(2)}=={\frac {e\hbar }{2imc}}\left\{{\boldsymbol {X}}^{(2)}\times {\boldsymbol {\mathcal {H}}}+{\frac {1}{2c}}{\boldsymbol {E}}\cdot {\boldsymbol {v}}{\boldsymbol {X}}^{(2)}-{\boldsymbol {\mathcal {H}}}X^{(0)}\right\},$

where

${\boldsymbol {\mathcal {H}}}={\boldsymbol {H}}+{\frac {1}{2c}}{\boldsymbol {E}}\times {\boldsymbol {v}},$

and ${\boldsymbol {v}}={\frac {\hbar }{im}}\nabla$ denotes the velocity operator.

Equations (10) are essentially equivalent to Darwin's equations.^[12] The only difference is in the terms

${\frac {e\hbar }{4imc^{2}}}{\boldsymbol {E}}\cdot {\boldsymbol {v}}={\frac {\hbar }{4ic^{2}}}{\frac {d{\boldsymbol {v}}}{dt}}{\boldsymbol {v}}={\frac {\hbar }{8ic^{2}}}{\frac {d{\boldsymbol {v}}^{2}}{dt}},$

which are obviously omitted in the averaging. Obtaining the experimentally confirmed equations can be taken as justification of our basic assumptions. We would like to emphasize once again that all the coefficients so much discussed resulted automatically without any special additional hypothesis. The rotation is completely replaced by the R-effect. According to (9), the density expression obtained from (3) is

(11)$\rho ={\frac {1}{c}}j_{0}={\frac {1}{2}}\sum _{N}(-1)^{N}\left\{X_{ab...}^{(N)}{\overline {\Theta }}_{ab...}^{(N)}+{\overline {X}}_{ab...}^{(N)}\Theta _{ab...}^{(N)}\right\}$

or since

${\boldsymbol {X}}^{(2N+1)}={\boldsymbol {\Theta }}^{(2N+1)}=0,$

${\boldsymbol {X}}^{(2N)}={\boldsymbol {\Theta }}^{(2N)},$

approximately, we can write

(12)$\rho =\sum _{N}X_{ab...}^{(2N)}{\overline {X}}_{ab...}^{(2N)}.$

This expression takes the place of the usual $\psi {\overline {\psi }}$.

The generalization of (9) to the multi-electron problem is now done without further ado. In order to give the equations the invariant form again, we want to introduce a new metric in our coordinate space, analogous to the relativistic treatment. Since, as already indicated, we are not dealing with an electronic property, but with a peculiar effect relating to all degrees of freedom, we must start with the expression for the Hamiltonian function, regardless of the question of the meaning of the momenta. In mechanics it is shown that in most cases the Hamiltonian function can be represented in the form

(13)${\frac {1}{2}}\sum \sum \gamma ^{ab}(p_{a}-f_{a})(p_{b}-f_{b})+V$

where $V$ and $f$ depend only on the coordinates. We choose the tensor $\gamma ^{ab}$ as the basic contravariant tensor. Then the R-equations (9) are written as

(14)$\left\{{\frac {\hbar }{i}}{\frac {\partial }{\partial t}}-{\frac {\hbar ^{2}}{2}}\Delta -{\frac {\hbar }{i}}f^{a}\nabla _{a}+{\frac {1}{2}}f^{a}f_{a}+V\right\}X_{ik...}^{(2N)}={\frac {\hbar }{2i}}\left\{M^{ab}X_{abik...}^{(2N+2)}-(M_{\alpha }^{a}X_{a\beta \gamma ...}^{(2N)})_{ik...}-{\frac {\hbar }{2ic^{2}}}\nabla ^{a}V\nabla _{a}X_{ik...}^{(2N)}-(M_{\alpha \beta }X_{\gamma \delta }^{(2N-1)})_{ik...}\right\},$

where

(14a)$M_{ik}=\nabla _{i}f_{k}-\nabla _{k}f_{i}+{\frac {\hbar }{2ic^{2}}}(\nabla _{k}V\nabla _{i}-\nabla _{i}V\nabla _{k}).$

$\Delta$ is not to be understood here as the sum of the second derivatives, but as the tensorial Laplacian operator.

In the many-electron problem, the kinetic energy is

(15)$T={\frac {1}{2}}\gamma ^{ab}p_{a}p_{b}={\frac {1}{2m}}\sum _{A}{\boldsymbol {p}}_{A}^{2}+{\frac {e^{2}}{m^{2}c^{2}}}\sum _{A}\sum _{B}{\boldsymbol {p}}_{A}{\boldsymbol {p_{B}}}$

where $A,\,B$ are the electron indices and ${\boldsymbol {p}}$s are the momenta. Because of the approximation used, what is important is only the change in $\Delta$ when acting on the tensor ${\boldsymbol {X}}^{(2N)}$ with respect to its effect on the scalar $X^{(0)}$, i.e.,

(16)$(\Delta -\Delta _{0})X_{ik...}^{(2N)}=2(\Gamma _{\alpha b}^{a}\nabla ^{b}X_{a\beta \gamma ...}^{(2N)})_{ik...}+((\nabla ^{b}\Gamma _{\alpha \beta }^{a})X_{a\beta \gamma ...}^{(2N)})_{ik...}$

where $\Gamma _{kl}^{i}$ is the three-indexed Christoffel symbols. These additional terms roughly correspond to the magnetic interaction of the electrons.

Because of the identity of all electrons, the question of how to formulate the Pauli principle still arises. As usual, he demands the "antisymmetry" of all wave functions. However, the change in indices must also be taken into account.

We hope to be able to give a closer look at the questions touched upon in a future work.

We are pleased to express our warmest thanks to our friend Prof J. Frenkel for the suggestion for this work and his constant supportive discussions. We would also like to sincerely thank Prof. V. Bursian for some valuable comments.

1. J. Frenkel, ZS. f. Phys., 37, 243, 1926.
2. L. H. Thomas, Phi. Mag., 3, 1, 1927.
3. GC. G. Darwin, Proc. Roy. Soc. (A) 116, 227, 1927.
4. P. J. Jordan, ZS. f. Phys., 44, 1, 1926.
5. P. A. M. Dirac, Proc. Roy. Sec. (A) 117, 610, 1928.
6. $\hbar$ denotes the Planck's constant divided by $2\pi$.
7. The overbar denotes the complex conjugate.
8. Like Dirac, we do not use higher powers of $u$.
9. The non-computing of identical terms should not be forgotten. For example,

   $(F_{\alpha \beta }\psi _{\gamma \delta }^{(2)})_{iklm}=F_{ik}\psi _{lm}^{(2)}+F_{il}\psi _{mk}^{(2)}+F_{im}\psi _{kl}^{(2)}+F_{lm}\psi _{ik}^{(2)}+F_{mk}\psi _{il}^{(2)}+F_{kl}\psi _{im}^{(2)}.$

10. $v$ denotes the order of magnitude of the velocity.
11. $\varphi$ denotes the scalar potential.
12. See C. G. Darwin, l.c., the vector introduced by him corresponds to $-i{\boldsymbol {X}}^{(2)}$.