Republication of: Dirac electron in the gravitational field I

Abstract

This paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series. It is one of the papers cited in N. D. Birrell and P. C. W. Davies’ book Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982) on the generalization of the Dirac Equation to a curved spacetime and is notable as containing the first occurrence of the well-known formula $g^{\mu \nu }\nabla _\mu \nabla _\nu +m^2+R/4$ for the ‘square’ of the Dirac operator. The original is in German and in a poorly accessible journal: it is presented here in English. The paper is accompanied by an editorial note by B. S. Kay which clarifies the historical development of the topic, and by a brief biography written by M. A. H. MacCallum.

Dirac electron in the gravitational field I

1 Introduction

The unification of the Dirac theory of the electron with the general theory of relativity has already been attempted repeatedly, for example by Wigner [1], Tetrode [2], Fock [3], Weyl [4], Zaycoff [5], Podolsky [6]. Most authors introduce in every world point axes of coordinates and numerically specialized Dirac matrices with respect to them. With this procedure it is a little bit difficult to recognize whether Einstein’s idea of teleparallelism, to which reference is partly made, really enters or whether one is independent of it. It is, moreover, necessary to recast the concepts of Riemannian geometry into the less familiar and definitely more complicated form of the “frame components”. In order to avoid all of this, it seemed to me desirable, like Tetrode (see also [7]), to rely only on the generalized commutation relations [see equation (2) below]. It turns out that one is led in this way very simply and straightforwardly to the important operators $\varGamma _k$, whose trace is the four-potential, and which Fock introduces as the “components of the parallel transport of a spinor”; and one is just as straightforwardly led to the important system of equations [see (8) below], which Fock obtains by a detour through the frame components. By restriction of the admissable reference frames (see Sec. 4 below), which is completely analogous to the usual one in the special theory of relativity, one then introduces the Hermiticities, which are desirable for the interpretation, as well as an assignment between tensor operators and local c-tensors, which is also completely analogous to the one put up by v. Neumann [8] in the special theory [see equation (57) below]. A principal advantage seems to me that the whole apparatus can be constructed almost entirely by pure operational calculus, without making reference to the $\psi$-function. I hope that the exact justification of this apparatus is not too shocking by its length, for which the author’s broad way of writing is partly responsible. Having prepared the ground, with it the application and the thinking may turn out to be simple. — I want to declare once and for all my deep indebtedness to the work of my predecessors, but for methodological reasons I ask for the permission to derive everything in a new way, as if it had not yet been found by anyone else.

2 Construction of the metric from fields of matrices

We call the world variables

$$\begin{aligned} x_0=ict, \ \ x_1=x, \ \ x_2=y, \ \ x_3=z. \end{aligned}$$

The first is always pure imaginary, the other three are real. Dirac’s basic idea was to interpret the Euclidean wave operator

$$\begin{aligned} \frac{\partial ^2}{\partial x_0^2}+ \frac{\partial ^2}{\partial x_1^2}+ \frac{\partial ^2}{\partial x_2^2}+ \frac{\partial ^2}{\partial x_3^2} \end{aligned}$$

as the square of a linear operator

$$\begin{aligned} \left( {\mathop {\gamma }\limits ^{0}}_0\frac{\partial }{\partial x_0}+ {\mathop {\gamma }\limits ^{0}}_1\frac{\partial }{\partial x_1}+ {\mathop {\gamma }\limits ^{0}}_2\frac{\partial }{\partial x_2}+ {\mathop {\gamma }\limits ^{0}}_3\frac{\partial }{\partial x_3}\right) ^2 \end{aligned}$$

where the ${\mathop {\gamma }\limits ^{0}}_k$ are $4\times 4$ matrices¹, which have to fulfil the condition

$$\begin{aligned} {\mathop {\gamma }\limits ^{0}}_i{\mathop {\gamma }\limits ^{0}}_k+{\mathop {\gamma }\limits ^{0}}_k{\mathop {\gamma }\limits ^{0}}_i=2\delta _{ik}, \end{aligned}$$

(1)

i.e. the left side is equal to the null matrix or equal to twice the identity matrix, depending on whether $i\ne k$ or $i=k$. By the condition (1), one knows that the ${\mathop {\gamma }\limits ^{0}}_k$ are exactly determined up to a similarity transformation

$$\begin{aligned} {\mathop {\gamma }\limits ^{0}}_k^{\ \prime }=S^{-1}{\mathop {\gamma }\limits ^{0}}_kS \end{aligned}$$

with an arbitrary non-singular $4\times 4$ transformation matrix S. This freedom in the choice of the ${\mathop {\gamma }\limits ^{0}}_k$ is evident, and one knows, as said, that with it the freedom is exhausted.

Since instead of the wave operator one could have also started from the squared line element:

$$\begin{aligned} dx_0^2+dx_1^2+dx_2^2+dx_3^2, \end{aligned}$$

it seems reasonable to interpret the conditions (1) in such a way that the matrices ${\mathop {\gamma }\limits ^{0}}_k$, in addition to the other tasks which they are attributed later in the description of the electron, also have the task of describing the world metric, which so far has been assumed Euclidean. If this is not assumed, but instead

$$\begin{aligned} ds^2=g_{\mu \nu }dx^{\mu }dx^{\nu }, \end{aligned}$$

one will have to replace (1) by

$$\begin{aligned} \gamma _i\gamma _k+\gamma _k\gamma _i=2g_{ik} \end{aligned}$$

(2)

[2]. The $\gamma _k$ are functions of space and time, i.e. they are $4\times 4$ matrices, whose elements are functions of the $x_i$.

In every point P, the equations (2) certainly have solutions for the $\gamma _k$ if one imagines the $g_{ik}$ somehow as given (of course in such a way that they correspond to a non-singular metric). The freedom which still exists for the $\gamma _k$, given the $g_{ik}$, is exactly the same as above for the ${\mathop {\gamma }\limits ^{0}}_k$, namely: transformation with an arbitrary non-singular matrix S. One recognizes the correctness of these statements by considering one by one the following:

1. The equations (2) can always be solved by 4 suitably chosen linear combinations of an arbitrary Dirac basis system ${\mathop {\gamma }\limits ^{0}}_k$ — the ansatz leads to conditions for the coefficients that can be met.

2. On the other hand: If one has a system of $\gamma _k$ for which one knows that it fulfils (2), one can specify 4 linear combinations of these $\gamma _k$ which fulfil (1) and which thus form a Dirac basis. If one thus has, for example, two systems of solutions $\gamma _k$ and $\gamma _k'$ for (2), they can be transformed into a respective Dirac basis by the same linear transformation. But these two Dirac bases are certainly related by an S-transformation. The same transformation then also transforms $\gamma _k$ and $\gamma _k'$ into each other.

3. It is obvious that any S-transformation leaves (2) untouched. — With this, all statements are demonstrated.

A very essential difference between the ${\mathop {\gamma }\limits ^{0}}_k$ and the $\gamma _k$ is the following. It is known that there are Hermitian ${\mathop {\gamma }\limits ^{0}}_k$-systems, but that there are in general no Hermitian $\gamma _k$-systems; there are also none where some $\gamma _k$ are Hermitian and others are skew-Hermitian. This is connected with the well known reality conditions which one has to demand for the $g_{ik}$: pure imaginary if one and only one index 0 appears, and real otherwise. (One has to recall that the symmetrized product, the anticommutator, of two Hermitian matrices is always Hermitian.) We will later address the Hermiticity questions in more detail and have mentioned them here only to show that for the moment there is not the slightest reason to restrict the transformation S, which is arbitrary in each point, for example to a unitary one. Because the $\gamma _k$ are anyway not Hermitian, one has for now no reason to be concerned about the “conservation of Hermiticity”.

We shall now derive from (2) an important system of differential equations for the $\gamma _k$. We imagine the $g_{ik}$ as given and the equations (2) as solved in every point P; solved in such a way that these solutions can be joined together to form four continuous, differentiable fields of matrices, which will obviously be possible. We now proceed from a point P to a neighbouring point $P'$ and form in this sense the complete differential of equation (2),

$$\begin{aligned} \delta \gamma _i\cdot \gamma _k+\gamma _i\cdot \delta \gamma _k +\delta \gamma _k\cdot \gamma _i+\gamma _k\cdot \delta \gamma _i= 2\frac{\partial g_{ik}}{\partial x_l}\delta x^l. \end{aligned}$$

(3)

If we now observe the theorem of Ricci, according to which the covariant derivative of the fundamental tensor $g_{ik}$ vanishes identically:

$$\begin{aligned} g_{ik;l}\equiv \frac{\partial g_{ik}}{\partial x_l}-\varGamma ^{\mu }_{kl}g_{i\mu } -\varGamma ^{\mu }_{il}g_{\mu k}\equiv 0, \end{aligned}$$

(4)

the right-hand side of (3) will be equal to the following:

$$\begin{aligned} 2\left( \varGamma ^{\mu }_{kl}g_{i\mu }+\varGamma ^{\mu }_{il}g_{\mu k}\right) \delta x^l. \end{aligned}$$

This value can be attributed to the left-hand side of (3) by setting

$$\begin{aligned} \delta \gamma _i=\varGamma ^{\mu }_{il}\gamma _{\mu }\delta x^l \end{aligned}$$

(5)

and taking into account (2). I.e. the matrices

$$\begin{aligned} \gamma _i+\delta \gamma _i=\gamma _i+\varGamma ^{\mu }_{il}\gamma _{\mu }\delta x^l \end{aligned}$$

(6)

obey equation (2) in point $P'$ if the $\gamma _i$ obey it in point P.

The ansatz (5) would in general be contradictory if one wanted to apply it to all points $P'$ in the neighbourhood of P. For one can convince oneself by a simple calculation that the expression (5) is a total differential if and only if the curvature in P vanishes. But according to what has been said above, the $\gamma _i$-values in $P'$ — we want to call them $\gamma _i+\delta '\gamma _i$ — can and will differ from our somehow guessed solution ansatz (5) resp. (6) by a similarity transformation, namely, of course, by an infinitely small one if continuity has to be preserved. That is, there must exist an infinitely small matrix $\epsilon$ in such a way that

$$\begin{aligned} \gamma _i+\delta '\gamma _i= & {} (1-\epsilon )(\gamma _i+\delta \gamma _i)(1+\epsilon ) = \gamma _i+\delta \gamma _i+\gamma _i\epsilon -\epsilon \gamma _i\nonumber \\ \text {or} \ \ \ \quad \delta '\gamma _i= & {} \varGamma ^{\mu }_{il}\gamma _{\mu }\delta x^l+\gamma _i\epsilon -\epsilon \gamma _i . \end{aligned}$$

(7)

In principle, $\epsilon$ could assume another, entirely arbitrary, value at any neighbouring point. But if $\gamma _i$ should have a correct differential quotient with respect to $x_l$ when progressing in direction $x_l$ (i.e. for $\delta x_l\ne 0$, all others $=0$), $\epsilon$ must be proportional to $\delta x_l$. The same for any l. If one should then be able to calculate the change of $\gamma _i$ when progressing in arbitrary direction correctly from its differential quotient, $\epsilon$ must be the sum of these four terms. In this way, one arrives at the ansatz

$$\begin{aligned} \epsilon =-\varGamma _l\delta x^l, \end{aligned}$$

in which the $\varGamma _i$ are four matrices independent of space and time (the minus sign is, of course, completely arbitrary). Inserted into (7), the important system of differential equations announced above follows²:

$$\begin{aligned} \frac{\partial \gamma _i}{\partial x_l}=\varGamma ^{\mu }_{il}\gamma _{\mu } +\varGamma _l\gamma _i-\gamma _i\varGamma _l. \end{aligned}$$

(8)

We shall express this later as follows: the covariant derivative of the fundamental vectors $\gamma _k$ vanishes, in full analogy to the theorem of Ricci, equation (4). On the other hand, the source freedom of the four-current is closely related to this system of equations. I want to put particular emphasis on the fact that we have derived it here purely from the conditions on the metric, without reference to the $\psi$-function, for which we had to exploit the freedom in transforming the Dirac matrices. This led — and it did it unavoidably — to the appearance of the new operators $\varGamma _l$, for which we shall see that they are inextricably linked with the four-potential (but they do not form a vector!).

We now investigate, in addition, the necessary conditions for the consistency of the equations (8), namely, that the mixed second differential quotients, when calculated in two different ways, must coincide. By expressing the first derivatives that appear after differentiation again by (8), one finds:

$$\begin{aligned} \varPhi _{kl}\gamma _i-\gamma _i\varPhi _{kl}= R^{...\mu }_{kli}\gamma _{\mu }. \end{aligned}$$

(9)

Here, $R^{...\mu }_{kli}$ is the mixed Riemann curvature tensor in the usual notation (see e.g. Levi-Civita, Der absolute Differentialkalkül, p. 91; Berlin, Springer 1928). $\varPhi _{kl}$ is an abbreviation that we introduce for the following six matrices, which are antisymmetric in the indices k, l:

$$\begin{aligned} \varPhi _{kl}=\frac{\partial \varGamma _l}{\partial x_k}-\frac{\partial \varGamma _k}{\partial x_l}+\varGamma _l\varGamma _k-\varGamma _k\varGamma _l, \end{aligned}$$

(10)

which, as it will turn out, stand in close relation to the electromagnetic field. For given $\gamma _i$-field, by (8) every $\varGamma _i$, and by (9) every $\varPhi _{kl}$ is fixed up to an added term which is commutable with all $\gamma _i$ and which is thus a multiple of the identity matrix. From (9), the $\varPhi _{kl}$ are easily calculable. Besides the $\gamma _i$ one introduces the contravariant

$$\begin{aligned} \gamma ^i=g^{ik}\gamma _k. \end{aligned}$$

(11)

Furthermore, one declares

$$\begin{aligned} s^{\mu \nu }=\frac{1}{2}\left( \gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu }\right) . \end{aligned}$$

(12)

(The $s^{\mu \nu }$ correspond for $\mu ,\nu =1,2,3$ in a certain sense to the spin, for $\mu =0, \nu =1,2,3$ in a certain sense to the velocity. See later.) We remark, in addition, that according to (2) and (11)

$$\begin{aligned} \gamma _i\gamma ^k+\gamma ^k\gamma _i=2\delta ^k_i. \end{aligned}$$

(13)

One now easily finds

$$\begin{aligned} \gamma _is^{\mu \nu }-s^{\mu \nu }\gamma _i=2\left( \delta ^{\mu }_i\gamma ^{\nu }- \delta ^{\nu }_i\gamma ^{\mu }\right) . \end{aligned}$$

(14)

Upon commutation with a $\gamma$, the $s^{\mu \nu }$ thus produce again $\gamma$. This is exactly what one needs in order to solve (9) with respect to $\varPhi _{kl}$. The right-hand side of (9) can, in fact, also be written as $R_{kl,i\mu }\gamma ^{\mu }$, where $R_{kl,i\mu }$ is the symmetric Riemann tensor. One then confirms with the C.R. (14) that

$$\begin{aligned} \varPhi _{kl}=-\frac{1}{4}R_{kl,\mu \nu }s^{\mu \nu }+f_{kl}\cdot 1 \end{aligned}$$

(15)

is the general solution of (9)³. $f_{kl}$ is the undetermined multiplier of unity. The $f_{kl}$ will (multiplied by i) assume the role of the electromagnetic field. One recognizes that although the appearance of these quantities is suggested by the construction of the metric from matrices, it is exactly the $f_{kl}$ which are, for the time being, not determined by the $\gamma$-field, but left entirely free.

As commutators, the $s^{\mu \nu }$ have trace zero. Therefore,

$$\begin{aligned} \mathrm{trace}\ \varPhi _{kl}=f_{kl}\cdot \mathrm{trace}\ 1=4f_{kl}. \end{aligned}$$

According to (10) we have, on the other hand,

$$\begin{aligned} \mathrm{trace}\ \varPhi _{kl}=\frac{\partial }{\partial x_k}\left( \mathrm{trace}\ \varGamma _l\right) - \frac{\partial }{\partial x_l}\left( \mathrm{trace}\ \varGamma _k\right) , \end{aligned}$$

for differentiation and performance of trace are commutable and the commutator does not contribute to the trace. If one sets, for example,

$$\begin{aligned} \frac{1}{4}\mathrm{trace}\ \varGamma _l=\varphi _l, \end{aligned}$$

one gets

$$\begin{aligned} f_{kl}=\frac{\partial \varphi _l}{\partial x_k}-\frac{\partial \varphi _k}{\partial x_l}. \end{aligned}$$

(16)

The traces of the $\varGamma _l$ are the four-potential (apart from a factor i).

3 Transformation theory, first part

According to the fundamental principle of general relativity, a re-labelling of all points

$$\begin{aligned} x_k'=x_k'(x_0,x_1,x_2,x_3); \quad k=0,1,2,3 \end{aligned}$$

(17)

should not change the form of description. In doing so, the function $x_0'$ should only assume pure imaginary, $x_1',x_2',x_3'$ only assume real values, and the functional determinant should remain positive. We call this a point substitution. The $g_{ik}$ then transform as a covariant tensor of second rank.

As long as we impose for the $\gamma _i$ no other requirement than obeying the equations (2), the question how they have to be transformed under a point substitution cannot at all be answered unambiguously. For after as well as before the point substitution, a similarity transformation with a transformation matrix S that varies from point to point remains entirely free. But we can demand that the $\gamma _i$ are to be transformed as a covariant vector under a pure point substitution, under which (2) at any rate is preserved. One then has to demand the same for the $\varGamma _l$, so that (8) is preserved. For the commutator $\varGamma _l\gamma _i-\gamma _i\varGamma _l$ then transforms as a covariant tensor, which is also the case for the rest of the equation, namely,

$$\begin{aligned} \frac{\partial \gamma _i}{\partial x_l}-\varGamma ^{\mu }_{il}\gamma _{\mu } \end{aligned}$$

(18)

if $\gamma _i$ is substituted as a vector; for (18) is, after all, formally the covariant derivative of $\gamma _i$. The similarity transformations

$$\begin{aligned} \gamma _k'=S^{-1}\gamma _kS \end{aligned}$$

(19)

would then have to be considered as something by themselves, where, as one can easily convince oneself, the $\varGamma _l$ must be transformed as follows in order to preserve (8):

$$\begin{aligned} \varGamma _l'=S^{-1}\varGamma _lS-S^{-1}\frac{\partial S}{\partial x_l}, \end{aligned}$$

(20)

which is different from the $\gamma _k$. But one would find that after these determinations the following combination of terms, for which we want to introduce the symbol $\nabla _k$,

$$\begin{aligned} \nabla _k=\frac{\partial }{\partial x_k}-\varGamma _k, \end{aligned}$$

(21)

first — naturally – transforms as a covariant vector under a pure point substitution (because this holds, of course, for the $\frac{\partial }{\partial x_k}$ alone and was fixed for the $\varGamma _k$) and that secondly because of (20) the $\nabla _k$ transform under an S-transformation exactly in the way that the $\gamma _k$ transform according to (19),

$$\begin{aligned} \nabla _k'=S^{-1}\nabla _kS. \end{aligned}$$

(22)

For the meaning of $\nabla _k'$ is, in fact,⁴

$$\begin{aligned} \nabla _k'= \frac{\partial }{\partial x_k}-\varGamma _k'= \frac{\partial }{\partial x_k}-S^{-1}\varGamma _kS+S^{-1}\frac{\partial S}{\partial x_k}, \end{aligned}$$

(23)

and one has

$$\begin{aligned} \frac{\partial }{\partial x_k}=\frac{\partial }{\partial x_k}\cdot S^{-1}S= S^{-1}\frac{\partial }{\partial x_k}S+\frac{\partial S^{-1}}{\partial x_k}S=S^{-1}\frac{\partial }{\partial x_k}S-S^{-1}\frac{\partial S}{\partial x_k}; \end{aligned}$$

(24)

the last equality holds because of the identity:

$$\begin{aligned} S^{-1}S\equiv 1; \quad \frac{\partial S^{-1}}{\partial x_k}S+S^{-1} \frac{\partial S}{\partial x_k}\equiv 0. \end{aligned}$$

By inserting (24) into (23), one confirms (22).

The $\varPhi _{kl}$ introduced through (10) would firstly behave — naturally — as a covariant tensor under point substitution, secondly under an S-transformation analogously to (19),

$$\begin{aligned} \varPhi _{kl}'=S^{-1}\varPhi _{kl}S, \end{aligned}$$

(25)

the latter because of (22) and because they are, according to the definitions (10) and (21), the commutators of the $\nabla _k$:

$$\begin{aligned} \varPhi _{kl}=\nabla _l\nabla _k-\nabla _k\nabla _l. \end{aligned}$$

(26)

One should add that due to (25) the traces of the $\varPhi _{kl}$, the $f_{kl}$, do not change under the similarity transformation, whereas those of the $\varGamma _l$, which we called $\varphi _l$, in fact do, because we do not have for the $\varGamma _l$ a transformation law analogous to (19) resp. (25), but instead (20).

We have formulated all of this in the “would”-form, because the requirements we have imposed contain the above mentioned arbitrariness: since a point substitution anyhow forces, in general, a modification of the $\gamma _i$ (the old $\gamma _i$ will, of course, in general no longer obey the equations (2)!), we have for the new selection again a whole manifold of $\gamma _i$-fields at our disposal, whose members follow from an arbitrary one of them by arbitrary, coordinate-dependent S-transformations. And for the moment none of these members is intrinsically distinguished in any way, not even the one selected above.

It is now recommendable, at least for certain purposes, to restrict this freedom of choice to a large extent by using it to satisfy certain Hermiticity aspirations, which are not unavoidable, but which are natural, as is also usually done in the special relativistic Dirac theory. In order to see what can be achieved in this respect, we have to contemplate more closely the eigenvalues of $\gamma _k$ and their bi-products.

4 Eigenvalues and Hermitization

Since according to (2)

$$\begin{aligned} \gamma _k\gamma _k=g_{kk}, \quad \text {(no summation!)} \end{aligned}$$

$\gamma _k$ has the eigenvalues $\pm \sqrt{g_{kk}}$, and each of these occur twice, because it has trace zero. The latter is seen if one sets analogously to (12)

$$\begin{aligned} s_{\mu \nu }=\frac{1}{2}\left( \gamma _{\mu }\gamma _{\nu }-\gamma _{\nu }\gamma _{\mu }\right) . \end{aligned}$$

Then one has in analogy to (14)

$$\begin{aligned} \gamma ^is_{\mu \nu }-s_{\mu \nu }\gamma ^i=2\left( \delta ^i_{\mu }\gamma _{\nu }-\delta ^i_{\nu } \gamma _{\mu }\right) . \end{aligned}$$

(27)

Every $\gamma$ can thus be represented in many ways as a commutator, and a commutator always has trace zero.

Although the $\gamma$ have only real eigenvalues and thus each single one of them can be made Hermitian by an S-transformation, this can, for example, in general not be done simultaneously, because according to (2) their symmetric product is equal to $2g_{01}\cdot 1$ and thus is (since $g_{01}$ is pure imaginary) skew-Hermitian.

Let us consider further the products $\gamma _i\gamma ^k$, first for $i\ne k$. Their square is [cf. (13)]:

$$\begin{aligned} \left( \gamma _i\gamma ^k\right) ^2=\gamma _i\gamma ^k\cdot \gamma _i\gamma ^k= -\gamma _i\gamma _i\gamma ^k\gamma ^k=-g_{ii}g^{kk}. \quad \text {(no summation!)} \end{aligned}$$

Thus the eigenvalues are $\pm i\sqrt{g_{ii}g^{kk}}$, and each of them occurs twice, since after all

$$\begin{aligned} \gamma _i\gamma ^k=\frac{1}{2}\left( \gamma _i\gamma ^k-\gamma ^k\gamma _i\right) \end{aligned}$$

as a commutator must have trace zero. — The eigenvalues of $\gamma ^k\gamma _i$ are equal and opposite. — On the other hand, one has for $i=k$:

$$\begin{aligned} \left( \gamma _k\gamma ^k\right) ^2=\gamma _k\gamma ^k\gamma _k\gamma ^k= & {} \gamma _k\left( 2-\gamma _k\gamma ^k\right) \gamma ^k=2\gamma _k\gamma ^k-g_{kk}g^{kk} \quad \text {(no s.)}\\ \left( \gamma _k\gamma ^k-1\right) ^2= & {} 1-g_{kk}g^{kk}.\quad \text {(no s.)} \end{aligned}$$

$\gamma _k\gamma ^k-1$ thus has the eigenvalues $\pm \sqrt{1-g_{kk}g^{kk}}$, and since it can be written as a commutator:

$$\begin{aligned} \gamma _k\gamma ^k-1=\frac{1}{2}\left( \gamma _k\gamma ^k-\gamma ^k\gamma _k\right) , \quad \text {(no s.)} \end{aligned}$$

each of them occurs twice. $\gamma _k\gamma ^k$ thus has the eigenvalues

$$\begin{aligned} 1\pm \sqrt{1-g_{kk}g^{kk}}, \end{aligned}$$

and each of them occurs twice. For $k=0$, these values are real, since $g_{00}g^{00}\le 1$.

Among the 4 matrices

$$\begin{aligned} \gamma _0\gamma ^0,\quad \gamma _0\gamma ^1,\quad \gamma _0\gamma ^2,\quad \gamma _0\gamma ^3,\quad \end{aligned}$$

(28)

the first one thus has only real, the three others pure imaginary eigenvalues. They thus just have (apart from a factor i) the reality conditions of a physically reasonable four-vector⁵. It thus seems reasonable to find out whether these four matrices can be Hermitized (resp. skew-Hermitized) simultaneously. It turns out that this is possible and that in addition a number of other matrices can be simultaneously Hermitized. This goes as follows.

If the metric $g_{ik}$ is real and positive definite, the equations (2) can be satisfied by Hermitian $\gamma _k$, in the same way as the equations (1) can be satisfied by Hermitian ${\mathop {\gamma }\limits ^{0}}_k$. This I may assume to be known without proof, the only thing involved being the projection of a ${\mathop {\gamma }\limits ^{0}}_k$-system, which is assumed Hermitian, from rectangular to skew coordinate axes, in which only real coefficients appear as direction cosines. And since the $g_{ik}$ are real in this case, the contravariant $\gamma ^k$ also turn out to be Hermitian; that is, one can also satisfy the contravariant analogues to (2),

$$\begin{aligned} \gamma ^i\gamma ^k+\gamma ^k\gamma ^i=2g^{ik}, \end{aligned}$$

(29)

by Hermitian $\gamma ^k$ if the tensor $g^{ik}$ is real and positive definite. This, however, is not the case for our tensor $g^{ik}$: one can make it the case if one mutilates $g^{ik}$ and simply neglects the “mixed” space-time components $g^{0k}$ for the time being, i.e. sets them to zero. Let

$$\begin{aligned} \alpha ^0, \alpha ^1, \alpha ^2, \alpha ^3 \end{aligned}$$

(30)

be a Hermitian quadruple of matrices which satifies the equations (29) with the mutilated metric. That is, one demands

$$\begin{aligned} \alpha ^i\alpha ^k+\alpha ^k\alpha ^i=2g^{ik} \end{aligned}$$

(31)

if neither or both indices i, k are equal to zero, and one demands for $k\ne 0$,

$$\begin{aligned} \alpha ^0\alpha ^k+\alpha ^k\alpha ^0=0. \end{aligned}$$

(32)

Let us now set

$$\begin{aligned} \gamma ^k=\frac{i}{g^{00}}\alpha ^0\alpha ^k \quad \text {for}\ k\ne 0 \end{aligned}$$

(33)

and

$$\begin{aligned} \gamma ^0=\frac{\alpha ^0}{\sqrt{g_{00}g^{00}}}-\frac{1}{g_{00}} \left( g_{01}\gamma ^1+g_{02}\gamma ^2+g_{03}\gamma ^3\right) . \end{aligned}$$

(34)

One can convince oneself by calculation that these $\gamma ^k$ obey the unmutilated equations (29).

Since according to (32) $\alpha ^0$ anticommutes with $\alpha ^k$ ($k\ne 0$), $\alpha ^0\alpha ^k$ is skew for $k\ne 0$ and thus $\gamma ^1,\gamma ^2,\gamma ^3$ are Hermitian according to (33). Furthermore, one calculates from (34)

$$\begin{aligned} \gamma _0=g_{0k}\gamma ^k=\alpha ^0\sqrt{\frac{g_{00}}{g^{00}}}=\text {Hermitian}. \end{aligned}$$

(35)

By our construction of the contravariant $\gamma ^1,\gamma ^2,\gamma ^3$ we thus have rendered Hermitian at the same time the covariant $\gamma _0$. — We note also the following Hermiticities: the contravariant purely spatial

$$\begin{aligned} s^{kl}=\frac{1}{2}\left( \gamma ^k\gamma ^l-\gamma ^l\gamma ^k\right) \quad \text {for}\ k,l=1,2,3 \end{aligned}$$

(36)

are, as commutators of Hermitian matrices, skew-symmetric. Furthermore, for $k\ne 0$ the $\gamma _0\gamma ^k$ and also the $\gamma ^k\gamma _0$ are skew, because, already according to (13), $\gamma _0$ anticommutes with $\gamma ^k$ ($k\ne 0$). One then finds from (34) and (35) that $\gamma _0\gamma ^0$ and $\gamma ^0\gamma _0$ are Hermitian. Furthermore, from this it follows very simply by lowering the index that for $k\ne 0$ also $\gamma _0\gamma _k$ and $\gamma _k\gamma _0$, and thus also

$$\begin{aligned} s_{0k}=\frac{1}{2}\left( \gamma _0\gamma _k-\gamma _k\gamma _0\right) \end{aligned}$$

turn out to be skew. But let us emphasize strongly that nothing can be said about the covariant $s_{kl}$ for $k,l\ne 0$ and also not about the contravariant $s^{0k}$! The same for $\gamma ^0,\gamma _1,\gamma _2,\gamma _3$. We summarize once again all statements. According to our construction,

$$\begin{aligned}&\gamma _0,\gamma ^1,\gamma ^2,\gamma ^3, \gamma _0\gamma ^0,\gamma ^0\gamma _0\quad \text {are Hermitian};\nonumber \\&\gamma _0\gamma _k,\gamma _k\gamma _0,\gamma _0\gamma ^k,\gamma ^k\gamma _0, s_{0k}, s^{kl} \quad \text {are skew}\quad (k,l\ne 0). \end{aligned}$$

(37)

We now finally want to liberate ourselves from the reference to a particular matrix construction, which has only served as an existence proof. One can easily understand the following: already the requirement that four suitably chosen matrices from the ones presented in (37) have the properties stated there — for example, the requirement that $\gamma _0,\gamma ^1,\gamma ^2,\gamma ^3$ be Hermitian — suffices to fix the $\gamma$-field for given $g_{ik}$ uniquely up to a unitary transformation. For there is no more freedom at all, given $g_{ik}$, for the $\gamma$-field than the following: transformation with an arbitrary matrix. If this transformation is supposed to leave the matrices $\gamma _0,\gamma ^1,\gamma ^2,\gamma ^3$ Hermitian, from which every matrix, that is, also every Hermitian matrix can be constructed by addition and multiplication⁶, the transformation must leave every Hermitian matrix Hermitian, i.e., it must be unitary. Q.E.D.

From now on we want to admit only such $\gamma$-fields — one could also say, only such reference frames —, for which the matrices $\gamma _0,\gamma ^1,\gamma ^2,\gamma ^3$ turn out to be Hermitian. All statements made in (37) then hold automatically. The “admissable” reference system is determined by the metric up to a unitary transformation.

It is very comfortable to have reduced the permitted S-transformations by this new requirement to unitary ones, for these are very good-natured and harmless. In general, we do not need to think of them and can proceed as if the $\gamma$-field was uniquely determined by the metric. But now, of course, the task arises of determining, when starting from an admissable $\gamma$-field and performing a point substitution (17), the transformation law of the $\gamma$ more specifically, namely determining it in such a way that one is led again to an admissable $\gamma$-field. The preliminary rule given at the beginning of section 3: to substitute the $\gamma _k$ as a covariant vector — does not at all obey this condition and does not, of course, correspond to how one proceeds in special relativity, where one does not at all substitute the ${\mathop {\gamma }\limits ^{0}}_k$. In the spirit of section 3 one could say: with every point substitution one must connect a fully determined (strictly speaking, determined up to a unitary factor!) S-transformation, which itself, of course, will not be unitary, and it is this transformation that has to be determined. One can thus rightfully speak of a complemented point substitution. In the next section, we shall perform this task for infinitely small point substitutions.

5 Transformation theory, second part

We start from an admissable $\gamma$-field and proceed to primed variables by the infinitely small point substitution

$$\begin{aligned} x_k'=x_k+\delta x_k\quad \text {or}\quad x_k=x_k'-\delta x_k, \end{aligned}$$

(38)

which we complement in the sense described above by an infinitely small S-transformation with

$$\begin{aligned} S=1+\varTheta ;\quad S^{-1}=1-\varTheta . \end{aligned}$$

(39)

We shall, as usual, not explicitly denote the change of the variables in the argument. The equations between primed and unprimed operators thus do not refer to the same, but to corresponding values of the argument, i.e. to the same point. — We now introduce the abbreviation

$$\begin{aligned} \frac{\partial \delta x_k}{\partial x_l}=a^k_l. \end{aligned}$$

(40)

These quantities are pure imaginary if one and only one index is equal to zero, and real otherwise. One then has

$$\begin{aligned} \gamma _i'= & {} \gamma _i-a^l_i\gamma _l+\gamma _i\varTheta -\varTheta \gamma _i\nonumber \\ \gamma '^{k}= & {} \gamma ^k+a^k_l\gamma ^l+\gamma ^k\varTheta -\varTheta \gamma ^k. \end{aligned}$$

(41)

If one takes the first equation for $i=0$ and multiplies it from the left into the second equation, one gets (always valid only in quantities of first order):

$$\begin{aligned} \gamma _0'\gamma '^k=\gamma _0\gamma ^k-a_0^l\gamma _l\gamma ^k+a_l^k\gamma _0\gamma ^l+ \gamma _0\gamma ^k\varTheta -\varTheta \gamma _0\gamma ^k. \end{aligned}$$

(42)

We use our freedom of choice for $\varTheta$ to eliminate resp. to replace the second term on the right-hand side of this equation, which prevents conclusions about Hermiticity. This can be achieved by

$$\begin{aligned} \varTheta =-\frac{1}{2g_{00}}a_0^l\gamma _l\gamma _0. \end{aligned}$$

(43)

For then one has

$$\begin{aligned} -2\varTheta \gamma _0\gamma ^k=a_0^l\gamma _l\gamma ^k, \end{aligned}$$

(44)

and one gets

$$\begin{aligned} \gamma _0'\gamma '^k=\gamma _0\gamma ^k+a_l^k\gamma _0\gamma ^l+ \gamma _0\gamma ^k\varTheta +\varTheta \gamma _0\gamma ^k. \end{aligned}$$

One can now convince oneself from our statements (37) that according to (43) $\varTheta$ is Hermitian. Its symmetrized product with $\gamma _0\gamma ^k$ is thus Hermitian or skew depending on whether $\gamma _0\gamma ^k$ has this property. The same holds for the second term on the right-hand side; it is skew for $k\ne 0$, Hermitian for $k=0$. Therefore the $\gamma _0'\gamma '^k$ keep the same Hermiticity as the $\gamma _0\gamma ^k$. One can show in the same way that $\gamma _0'$ stays Hermitian, too. With that, the $\gamma '$-field is legitimized as “admissable”.

$\varTheta$ is, of course, not unique, but the value given in (43) has, after all, this meaning: it is uniquely the Hermitian part of the infinitely small matrix to be used. There could be, in addition, an arbitrary infinitely small skew part. With some thought one recognizes that it would leave all conclusions unchanged – it corresponds, of course, only to an additional unitary transformation! —

We now add the exact definition of a tensor operator. If it is known or fixed that a system of operators

$$\begin{aligned} T^{\rho \sigma ..}_{\alpha \beta ..} \end{aligned}$$

transforms under an infinitely small complemented point substitution as a tensor according to the rank indicated by the indices and their positions, but with addition of the commutator

$$\begin{aligned} T^{\rho \sigma ..}_{\alpha \beta ..}\varTheta -\varTheta T^{\rho \sigma ..}_{\alpha \beta ..}, \end{aligned}$$

(45)

we want to call the system of operators a tensor operator of the corresponding rank.

The following important theorem⁷ then holds, which can be found by very easy generalizations of the above conclusions:

Let $T^{\rho \sigma ..}_{\alpha \beta ..}$ be a tensor operator and let it be known that in one reference frame the operators

$$\begin{aligned} \gamma _0T^{\rho \sigma ..}_{\alpha \beta ..} \end{aligned}$$

(46)

are Hermitian or skew, depending on whether zero occurs in the indices $\alpha \beta \cdot \cdot \rho \sigma \cdot \cdot$ in even or odd multiples; then this fact remains true in every reference frame. — In this theorem one may, of course, also exchange the words even and odd, i.e. one can take into account the zero in $\gamma _0$ or not. But what one must not do is concern oneself with the Hermiticity of $T^{\rho \sigma ..}_{\alpha \beta ..}$ itself, which is completely irrelevant; it is the one of $\gamma _0T^{\rho \sigma ..}_{\alpha \beta ..}$ that is relevant! —

One easily confirms that the symbol

$$\begin{aligned} \nabla _k=\frac{\partial }{\partial x_k}-\varGamma _k \end{aligned}$$

introduced in (21) is a vector operator. $\varGamma _k$ by itself is not, it transforms [with consideration of (20)] under a complemented point substitution obviously as:

$$\begin{aligned} \varGamma _k'=\varGamma _k-a^i_k\varGamma _i+ \varGamma _k\varTheta -\varTheta \varGamma _k-\frac{\partial \varTheta }{\partial x_k}. \end{aligned}$$

(47)

Here, the last term is surplus, being in conflict with the vector property. The pure differentiator $\frac{\partial }{\partial x_k}$, on the other hand, transforms covariantly in the elementary sense, without the $\varTheta$-commutator. Taking the two together, these evils compensate, because $\frac{\partial \varTheta }{\partial x_k}$ can be interpreted as a commutator of $\frac{\partial }{\partial x_k}$ and $\varTheta$. — To talk about “Hermitian” or “skew” makes, of course, no immediate sense for operators such as $\nabla _k$ which contain differentiations. For this reason, we have also not included this in the definition of a tensor operator.

If one has two tensor operators, one easily confirms by multiplication of their transformation formulae [similarly to what was done above in the transition from (41) to (42)], that one obtains by “writing next to each other”, i.e. matrix multiplication, again a tensor if the operator written on the left side does not contain the differential operator. Otherwise not, because it is then not commutable with the substitution coefficients $a^k_l$. (This is, of course, also not different in standard tensor calculus. Although there $\frac{\partial }{\partial x_k}$ is a vector, one does not, after all, obtain a tensor by the usual differentiation of tensor components, but by covariant differentiation.) The tensor character of the $\varPhi _{kl}$ defined by (10) or (26) must be investigated separately. But since we have already seen in section 3 that the $\varPhi _{kl}$ behave as a tensor under pure point substitution in the elementary sense referred to there, but transform under every S-substitution according to (25), they evidently form also a tensor operator under complemented point substitution in the finer sense considered now.

We now want to take care of what may be understood by covariant differentiation of a tensor operator. In this, we restrict ourselves to such operators which do not contain the differential operator, that is, to $4\times 4$ matrices whose elements are functions of coordinates (which does not prevent them having the form of differential quotients; for example, $\varPhi _{kl}$ is allowed, but $\nabla _k$ is not). The point is to derive from a tensor operator $T^{\rho \sigma ..}_{\alpha \beta ..}$ by differentiation with respect to $x_{\lambda }$ and addition of suitable complementary terms entities which transform under a complemented point substitution as a tensor operator that is contravariant in $\rho \sigma \cdot \cdot$ and covariant in $\alpha \beta \ldots \lambda$.

We make use of the fact that a complemented point substitution decomposes formally into a pure point substitution and a $\varTheta$-transformation, where in the latter one simply adds the commutator with $\varTheta$; we use, furthermore, that these two infinitely small transformations are, of course, commutable. Let us now consider the covariant differential quotient in the elementary sense,

$$\begin{aligned} \frac{\partial T^{\rho \sigma ..}_{\alpha \beta ..}}{\partial x_{\lambda }}- \varGamma ^{\mu }_{\alpha \lambda }T^{\rho \sigma ..}_{\mu \beta ..}-+\ldots , \end{aligned}$$

(48)

this will, of course, transform under a pure point substitution as a tensor of rank $^{\rho \sigma ..}_{\alpha \beta .. \lambda }$. It would only be necessary to show, in addition, that under a $\varTheta$-transformation it simply adds the commutator with $\varTheta$, as $T^{\rho \sigma ..}_{\alpha \beta ..}$ itself does. This is true for all terms in the expression stated before except for the first, in which by the $\varTheta$-transformation the term

$$\begin{aligned} \frac{\partial \left( T^{\rho \sigma ..}_{\alpha \beta ..}\varTheta -\varTheta T^{\rho \sigma ..}_{\alpha \beta ..}\right) }{\partial x_{\lambda }} \end{aligned}$$

is added instead of

$$\begin{aligned} \frac{\partial T^{\rho \sigma ..}_{\alpha \beta ..}}{\partial x_{\lambda }} -\varTheta \frac{\partial T^{\rho \sigma ..}_{\alpha \beta ..}}{\partial x_{\lambda }}. \end{aligned}$$

There thus emerges the surplus term

$$\begin{aligned} T^{\rho \sigma ..}_{\alpha \beta ..}\frac{\partial \varTheta }{\partial x_{\lambda }}-\frac{\partial \varTheta }{\partial x_{\lambda }}T^{\rho \sigma ..}_{\alpha \beta ..}. \end{aligned}$$

(49)

We remove it by adding in (48) as a completion the commutator

$$\begin{aligned} T^{\rho \sigma ..}_{\alpha \beta ..}\varGamma _{\lambda }-\varGamma _{\lambda } T^{\rho \sigma ..}_{\alpha \beta ..}. \end{aligned}$$

In this way, we arrive at the final definition for the covariant differentiation of a tensor operator:

$$\begin{aligned} T^{\rho \sigma ..}_{\alpha \beta ..;\lambda }= \frac{\partial T^{\rho \sigma ..}_{\alpha \beta ..}}{\partial x_{\lambda }} -\varGamma ^{\mu }_{\alpha \lambda }T^{\rho \sigma ..}_{\mu \beta ..}-+\ldots + T^{\rho \sigma ..}_{\alpha \beta ..}\varGamma _{\lambda }-\varGamma _{\lambda }T^{\rho \sigma ..}_{\alpha \beta ..}. \end{aligned}$$

(50)

Proof

According to (47), the added term behaves as follows: under a pure point transformation, as a tensor of the desired rank; under a $\varTheta$-transformation, it adds, firstly, its commutator with $\varTheta$ and, secondly, it removes the surplus (49). With this, the proof that (50) is a tensor is completed. One can write (50) also in the form:

$$\begin{aligned} T^{\rho \sigma ..}_{\alpha \beta ..;\lambda }=\nabla _{\lambda }T^{\rho \sigma ..}_{\alpha \beta ..} -T^{\rho \sigma ..}_{\alpha \beta ..}\nabla _{\lambda }-\varGamma ^{\mu }_{\alpha \lambda }T^{\rho \sigma ..}_{\mu \beta ..} -+, \end{aligned}$$

(51)

which differs from the elementary formula only by the appearance of the differentiator $\nabla _{\lambda }$ instead of the simple $\frac{\partial }{\partial x_k}$. $\square$

One now recognizes that the important system of differential equations (8), which we have met already at the beginning of our investigations, expresses nothing more than the vanishing of the covariant derivatives of the metric vector $\gamma _k$. This is in full analogy to the theorem of Ricci, which states the same for the metric tensor $g_{ik}$. Exactly the same holds, by the way, for every tensor derived from the $\gamma _k$ by multiplication and addition with constant coefficients, e.g. $\gamma ^k, s_{\mu \nu },s^{\mu \nu }$ etc. All of these have the covariant derivative zero. This is an immediate consequence of the equations (8).

6 Interpretation by the $\psi$-spinor

The restriction of the $\gamma$-fields to what we called “admissable” will be felt to be especially comfortable if the interpretation of the operators is based on a four-component $\psi$-function, a so-called spinor, on which they act. If a system of equations

$$\begin{aligned} T^{\rho \sigma ..}_{\alpha \beta ..}\psi =0 \end{aligned}$$

(52)

has to remain valid in any reference frame if it holds in one frame, one must demand that $\psi$, as an invariant of an S-transformation, transforms under a pure point substitution as follows:

$$\begin{aligned} \psi '=S^{-1}\psi . \end{aligned}$$

(53)

The first is self-evident. And during an S-transformation it follows indeed from (52) by multiplication from the left by $S^{-1}$ that

$$\begin{aligned} S^{-1}T^{\rho \sigma ..}_{\alpha \beta ..}SS^{-1}\psi =T'^{\rho \sigma ..}_{\alpha \beta ..} \psi '=0. \end{aligned}$$

For a complemented infinitely small point substitution one will thus have to set

$$\begin{aligned} \psi '=\psi -\varTheta \psi , \end{aligned}$$

(54)

where $\varTheta$ is the Hermitian matrix (43). Since now $\nabla _k$ is a vector operator, there follows among other things: if the four numbers

$$\begin{aligned} \nabla _k\psi =\frac{\partial \psi }{\partial x_k}-\varGamma _k\psi \end{aligned}$$

(55)

vanish in one reference frame, they do so in every frame. It is appropriate to call them covariant derivatives of the spinor $\psi$.

From the operators (q-numbers), one gets the ordinary numbers (c-numbers), which according to taste and mode of expression can be interpreted phyically as position probability, density of electricity, current density, transition probability etc., as follows: one applies the corresponding operator A to a spinor $\psi$: $A\psi$, and then forms the so-called Hermitian inner product of the two spinors $\psi$ and $A\psi$, i.e. one multiplies the first component of the conjugate complex $\psi ^*$ with the first of $A\psi$, the second of $\psi ^*$ with the second of $A\psi$ etc. and then adds these 4 products. For this, we want to write briefly⁸

$$\begin{aligned} \psi ^*A\psi . \end{aligned}$$

(56)

If A does not contain the differential operator $\frac{\partial }{\partial x_k}$, but is only a $4\times 4$ matrix with coordinate dependent elements, one can also say: one inserts the components of $\psi ^*$ and $\psi$ as arguments into the bilinear form constructed from this matrix.

Only if the matrix is Hermitian (resp. skew) will the c-number (56) always be real (resp. purely imaginary), as is necessary for the components of c-tensors which one wants to interpret physically. We have now seen in section 5: if $T^{\rho \sigma ..}_{\alpha \beta ..}$ is a tensor operator, under an admissable transformation (i.e. under a complemented point substitution) one will not at all preserve the Hermiticity of its components, but instead those of $\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..}$. The reality conditions needed for a physical tensor of rank $^{\rho \sigma ..}_{\alpha \beta ..}$ are thus not at all preserved by, for example, the c-numbers $\psi ^*T^{\rho \sigma ..}_{\alpha \beta ..}\psi$, but by the c-numbers

$$\begin{aligned} \mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..}=\psi ^*\gamma _0T^{\rho \sigma ..}_{\alpha \beta ..}\psi . \end{aligned}$$

(57)

We now want to show that it is also them that really transform as a c-tensor of rank $^{\rho \sigma ..}_{\alpha \beta ..}$ and thus have to be counted as the physical interpretation of the tensor operators $T^{\rho \sigma ..}_{\alpha \beta ..}$. This is because one finds, when performing the complemented point substitution (38), (40), first the following:

$$\begin{aligned} \mathrm{T}^{\rho \sigma ..\prime }_{\alpha \beta ..}= & {} (\psi ^*-\varTheta ^*\psi ^*) (\gamma _0-a_0^l\gamma _l+\gamma _0\varTheta -\varTheta \gamma _0)\nonumber \\&\quad (T^{\rho \sigma ..}_{\alpha \beta ..}-a_{\alpha }^lT^{\rho \sigma ..}_{l\beta ..}-+\ldots )(\psi -\varTheta \psi )\nonumber \\= & {} \mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..}-\varTheta ^*\psi ^*\gamma _0T^{\rho \sigma ..}_{\alpha \beta ..}\psi -\psi ^*\varTheta \gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..}\psi - a_0^l\psi ^*\gamma _l T^{\rho \sigma ..}_{\alpha \beta ..}\psi - \nonumber \\&\ \ \ \ \ \ \ \ -a_{\alpha }^l T^{\rho \sigma ..}_{l\beta ..}-+ \cdot \cdot \end{aligned}$$

(58)

(Two terms containing $\varTheta$ have cancelled each other, namely the one arising from $-\varTheta \psi$ and the one arising from $\gamma _0\varTheta$; terms of second order in $\varTheta$ and $a_k^l$ are, of course, suppressed.) The second, third, and fourth terms on the right-hand side cancel each other, for: the second and the third are equal to each other, because $\varTheta ^*$ may be transferred under “transposition” (exchange of rows and columns) to the other factor and in this way becomes $\varTheta$ because it is Hermitian. Furthermore, one has from (43)

$$\begin{aligned} -2\psi ^*\varTheta \gamma _0T^{\rho \sigma ..}_{\alpha \beta ..}\psi = \frac{1}{g_{00}}a_0^l\psi ^*\gamma _l\gamma _0\gamma _0T^{\rho \sigma ..}_{\alpha \beta ..}\psi = a_0^l\psi ^*\gamma _lT^{\rho \sigma ..}_{\alpha \beta ..}\psi , \end{aligned}$$

which thus cancels against the fourth term, as stated. One thus obtains for the c-tensor (57)

$$\begin{aligned} \mathrm{T}^{\rho \sigma ..\prime }_{\alpha \beta ..}=\mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..} -a_{\alpha }^l \mathrm{T}^{\rho \sigma ..}_{l\beta ..}-+\ldots , \end{aligned}$$

(59)

the usual substitution formula, Q.E.D. — One should note explicitly that in this proof the operator $T^{\rho \sigma ..}_{\alpha \beta ..}$ itself does not need to be moved nor to be commuted with a $a_k^l$. The proof thus also still holds, i.e. $\mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..}$ even then transforms as a c-tensor, if $T^{\rho \sigma ..}_{\alpha \beta ..}$ contains the differential operator $\frac{\partial }{\partial x_k}$. Only the Hermiticity statements then make no immediate sense for the local tensor components.

For the following it will be convenient to extend formula (55) to the case when one does not have a spinor, but instead its complex-conjugate. The complex-conjugate of (55) would read

$$\begin{aligned} \frac{\partial \psi ^*}{\partial x_k*}-\varGamma _k^*\psi ^*, \end{aligned}$$

but this would for $k=0$ ($x_0=ict$!) in the Euclidean case not become the usual derivative, but its negative, which would be very inconvenient. We are thus, unfortunately, forced to change the sign for $k=0$ and to define the covariant derivative of $\psi ^*$ as

$$\begin{aligned} \nabla _k\psi ^*=\frac{\partial \psi ^*}{\partial x_k}\mp \varGamma _k^*\psi ^* \end{aligned}$$

(60)

(upper sign for $k=1,2,3$; lower sign for $k=0$.) We now want, in addition, to investigate the covariant derivative of the c-tensor (57), which, as we expect, will be somehow connected with the one of the tensor operator defined in (50). One first finds:

$$\begin{aligned} \mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..;\lambda }= & {} \frac{\partial \mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..}}{\partial x_{\lambda }}- \varGamma ^{\mu }_{\lambda \alpha }\mathrm{T}^{\rho \sigma ..}_{\mu \beta ..}-+ \ldots =\\= & {} \frac{\partial \psi ^*}{\partial x_{\lambda }}\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..} \psi + \psi ^*\frac{\partial \gamma _0}{\partial x_{\lambda }}T^{\rho \sigma ..}_{\alpha \beta ..} \psi +\psi ^*\gamma _0 \frac{\partial T^{\rho \sigma ..}_{\alpha \beta ..}}{\partial x_{\lambda }}\psi +\psi ^*\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..} \frac{\partial \psi }{\partial x_{\lambda }}-\\- & {} \varGamma ^{\mu }_{\lambda \alpha }\psi ^*\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..}\psi -+ \ldots . \end{aligned}$$

One can now extend the four derivatives in this equation to covariant derivatives according to (60), (8), (50), (55), in which the one of $\gamma _0$ vanishes. In this way, one obtains

$$\begin{aligned} \mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..;\lambda }=\nabla _k\psi ^*\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..} \psi +\psi ^*\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..;\lambda }\psi +\psi ^*\gamma _0 \mathrm{T}^{\rho \sigma ..}_{\alpha \beta ..}\nabla \psi \end{aligned}$$

(61)

plus a remainder, for which it has to be shown now that it vanishes. This remainder is

$$\begin{aligned} \mathrm{remainder}= & {} \pm \varGamma _{\lambda }^*\psi ^*\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..}\psi + \\&+\psi ^*[\varGamma ^{\mu }_{0\lambda }\gamma _{\mu }T^{\rho \sigma ..}_{\alpha \beta ..} +(\varGamma _{\lambda }\gamma _0-\underline{\gamma _0\varGamma _{\lambda }})T^{\rho \sigma ..}_{\alpha \beta ..} - \underline{\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..} \varGamma _{\lambda } +\gamma _0\varGamma _{\lambda }T^{\rho \sigma ..}_{\alpha \beta ..}}]\psi +\\&+ \underline{\psi ^*\gamma _0 T^{\rho \sigma ..}_{\alpha \beta ..} \varGamma _{\lambda }\psi }. \end{aligned}$$

The underlined terms cancel each other. $\pm \varGamma _{\lambda }^*$ is transferred to the other factor as $\pm \varGamma _{\lambda }^{\dagger }$.⁹ There is still

$$\begin{aligned} \mathrm{remainder}= & {} \psi ^*A T^{\rho \sigma ..}_{\alpha \beta ..}\psi \ \mathrm{with} \\ A= & {} \varGamma ^{\mu }_{0\lambda }\gamma _{\mu }+(\varGamma _{\lambda }\pm \varGamma _{\lambda }^{\dagger })\gamma _0. \end{aligned}$$

The proof will be completed if we can show that

$$\begin{aligned} \frac{1}{2g_{00}}A\gamma _0 \equiv \frac{1}{2} (\varGamma _{\lambda }\pm \varGamma _{\lambda }^{\dagger }) + \frac{1}{2g_{00}}\varGamma ^{\mu }_{0\lambda }\gamma _{\mu }\gamma _0 \end{aligned}$$

(62)

vanishes. (For from this one has $A\equiv 0$, because $\gamma _0$ has the non-vanishing eigenvalues $\pm \sqrt{g_{00}}$. If $A=0$, the “remainder” vanishes and equation (61) will be proved.)

The operator (62) now is in the case of the upper sign, valid for $\lambda =1,2,3$, the Hermitian, in the case of the lower sign the skew-Hermitian part of

$$\begin{aligned} \varGamma _{\lambda }+\frac{1}{2g_{00}}\varGamma ^{\mu }_{0\lambda }\gamma _{\mu }\gamma _0. \end{aligned}$$

(63)

It can be recognized without too much effort that this operator, if commuted with the according to (37) Hermitian matrices $\gamma _0,\gamma ^1,\gamma ^2,\gamma ^3$ gives for $\lambda =1,2,3$ only Hermitian, for $\lambda =0$ only skew-Hermitian results. For this reason, its Hermitian (resp. for $\lambda =0$ its skew-Hermitian) part must at any rate be commutable with $\gamma _0,\gamma ^1,\gamma ^2,\gamma ^3$, thus must be a multiple of unity. In other words, the parts which are supposed to vanish reduce to

$$\begin{aligned} \mathrm{real\ part\ trace}\ \ (\varGamma _{\lambda }+ & {} \frac{1}{2g_{00}}\varGamma ^{\mu }_{0\lambda }\gamma _{\mu }\gamma _0) \ \ \mathrm{for}\ {\lambda =1,2,3} \\ \mathrm{imaginary\ part \ trace}\ \ (\varGamma _0+ & {} \frac{1}{2g_{00}}\varGamma ^{\mu }_{00}\gamma _{\mu }\gamma _0). \end{aligned}$$

Since trace $\gamma _{\mu }\gamma _0=4g_{\mu 0}$ and

$$\begin{aligned} g_{\mu 0}\varGamma ^{\mu }_{0\lambda }=\varGamma _{0,0\lambda }=\frac{1}{2}\frac{\partial g_{00}}{\partial x_{\lambda }} \ \ \mathrm{for}\ {\lambda =0,1,2,3} \end{aligned}$$

the question is thus whether one really has

$$\begin{aligned} \mathrm{real\ part\ trace}\ \ \varGamma _{\lambda }= & {} -\frac{\partial \lg g_{00}}{\partial x_{\lambda }} \ \ \mathrm{for}\ {\lambda =1,2,3}; \nonumber \\ \mathrm{imaginary\ part \ trace}\ \ \varGamma _{\lambda }= & {} -\frac{\partial \lg g_{00}}{\partial x_0}? \end{aligned}$$

(64)

It now turns out that we have promised too much. This is because we cannot prove these equations, the reason being that the $\varGamma _k$ were, after all, originally introduced and so far exclusively applied in such a manner that only their commutators with other matrices play a role, for which their traces are completely irrelevant. These play a role for the first time in the covariant derivative of the spinor, equation (55) and (60), of which we just make use for the first time in the equation (61) that we want to prove. We can only prove that we are free to define the corresponding trace parts by (64). And this is indeed the case. On the one hand, it certainly holds in one reference frame, because the right-hand sides of (64) possess the necessary reality. On the other hand, one can show from (47) and (43) that the decree once imposed is invariant under admissable transformations — I suppress the proof.

By this decree, the covariant derivative of the spinor is made precise in a desired way. But the decree is, in fact, desired also in another way. If the trace parts in question cannot be described as the derivatives of one and the same function $(-\lg g_{00})$, they would generate pure imaginary electromagnetic field strengths in the traces of the $\varPhi _{kl}$. This is avoided in this way. — The real part of trace $\varGamma _0$ and the imaginary parts of trace $\varGamma _{\lambda }$ ($\lambda =1,2,3$), from which the real field strength follow, still remain free.

We must, in addition, take a look at the pure unitary transformations which besides the complemented point substitutions are also still admissable in themselves. The only remaining thing to be said is that such a desired unitary transformation must, of course, also be applied to $\psi$ according to the prescription (53). It is then completely irrelevant and harmless. In particular, the components of the c-tensors (57) are completely insensitive to it; this also holds for the trace parts that were fixed in (64).

The essential results of this section are:

• 1. The determination of the transformation law (54) and the covariant derivative (55) for the spinor.

• 2. The assignment of the c-tensor components to the tensor operator according to (57) and the proof that they really transform as usual tensor components of the same rank.

• 3. The presentation of a relatively simple formula (61) for calculating the covariant derivative of a c-tensor; a formula which mainly is of interest because it demands for its validity the, in principle welcome,

• 4. normalization of that trace part of $\varGamma _{\lambda }$ that without normalization would lead to the appearance of pure imaginary electromagnetic field strengths.

7 The Dirac equation

The operator $\gamma ^k\nabla _k$ is an invariant which one can suitably call “absolute value of the gradient”. The generalized Dirac equation demands¹⁰

$$\begin{aligned} \gamma ^k\nabla _k=\mu \psi , \end{aligned}$$

(66)

where $\mu$ is a universal constant,

$$\begin{aligned} \mu =\frac{2\pi mc}{h}. \end{aligned}$$

Let us call the c-vector belonging to $\gamma ^k$ after the assignment (57) $iS^k$, that is,

$$\begin{aligned} iS^k=\psi ^*\gamma _0\gamma ^k\psi . \end{aligned}$$

(67)

Since the covariant derivative of the operator $\gamma ^k$ vanishes, the one of $S^k$ reduces according to (61) to¹¹

$$\begin{aligned} iS^k_{;\lambda }=\nabla _{\lambda }\psi ^*\gamma _0\gamma ^k\psi + \psi ^*\gamma _0\gamma ^k\nabla _{\lambda }\psi . \end{aligned}$$

If one forms by contraction the covariant divergence:

$$\begin{aligned} iS^{\lambda }_{;\lambda }=\nabla _{\lambda }\psi ^*\gamma _0\gamma ^{\lambda }\psi + \psi ^*\gamma _0\gamma ^{\lambda }\nabla _{\lambda }\psi , \end{aligned}$$

the first summand is the negative complex-conjugate of the second¹², but this one is according to (68)

$$\begin{aligned} \mu \psi ^*\gamma _0\psi , \end{aligned}$$

and thus is real, because $\gamma _0$ is Hermitian. Therefore,

$$\begin{aligned} S^{\lambda }_{;\lambda }=0. \end{aligned}$$

(70)

In this way, the source freedom of the four-current, which according to our assignment (57) belongs as a c-vector to the contravariant metric vector, follows from the Dirac equation and the fundamental equations (8) (cf. [3], p. 267).

We now want to square the Dirac equation in order to compare the result with the one familiar from the special theory (for brevity, $\psi$ will be suppressed):

$$\begin{aligned} \gamma ^k\nabla _k\gamma ^l\nabla _l=\mu ^2. \end{aligned}$$

(71)

One replaces the first two factors by equation (66) (in the footnote) and uses that one has according to (2) and (12)

$$\begin{aligned} \gamma ^k\gamma ^l=g^{kl}+s^{kl}. \end{aligned}$$

(72)

This leads to

$$\begin{aligned} \nabla _k(g^{kl}+s^{kl})\nabla _l+ \frac{\partial \lg \sqrt{g}}{\partial x_{\mu }}\gamma ^{\mu }\gamma ^l\nabla _l=\mu ^2. \end{aligned}$$

From the vanishing of the covariant derivative of $s^{kl}$ follows

$$\begin{aligned} \nabla _ks^{kl}-s^{kl}\nabla _k=-\frac{\partial \lg \sqrt{g}}{\partial x_{\mu }}s^{\mu l}. \end{aligned}$$

This leads to [after using again (72)]:

$$\begin{aligned} \nabla _kg^{kl}\nabla _l+s^{kl}\nabla _k\nabla _l+\frac{\partial \lg \sqrt{g}}{\partial x_{\mu }}g^{\mu l}\nabla _l=\mu ^2. \end{aligned}$$

For the second term one finds from (26) and because of the antisymmetry of the $s^{kl}$ that it is equal to $-\frac{1}{2}s^{kl}\varPhi _{kl}$. The first and the third (in which one replaces $\mu$ by k) combine to the generalized Laplace operator; one then finally gets:

$$\begin{aligned} \frac{1}{\sqrt{g}}\nabla _k\sqrt{g}g^{kl}\nabla _l- \frac{1}{2}s^{kl}\varPhi _{kl}=\mu ^2. \end{aligned}$$

(73)

It is of interest to insert here for $\varPhi _{kl}$ the expression (15) found much earlier. In this, the invariant

$$\begin{aligned} \frac{1}{8} R_{kl,\mu \nu }s^{kl}s^{\mu \nu } \end{aligned}$$

appears. Due to the symmetry of the covariant Riemann curvature tensor in the first and second index pair this is also equal to

$$\begin{aligned} \frac{1}{16}R_{kl,\mu \nu }(s^{kl}s^{\mu \nu }+s^{\mu \nu }s^{kl}). \end{aligned}$$

If one now — something that I do not want to carry out in extenso — really calculates the symmetrized products of the $s^{kl}$ and then makes use of the known cyclic symmetry

$$\begin{aligned} R_{kl,\mu \nu }+R_{l\mu ,k\nu }+R_{\mu k,l\nu }=0, \end{aligned}$$

one finally gets

$$\begin{aligned} \frac{1}{8} R_{kl,\mu \nu }s^{kl}s^{\mu \nu }=-\frac{1}{4}g^{k\mu }g^{l\nu }R_{kl,\mu \nu }=-\frac{R}{4}, \end{aligned}$$

where R is the invariant curvature. Consequently, inserting $\varPhi _{kl}$ from (15) into (73) gives the following:

$$\begin{aligned} \frac{1}{\sqrt{g}}\nabla _k\sqrt{g}g^{kl}\nabla _l-\frac{R}{4} -\frac{1}{2}f_{kl}s^{kl}=\mu ^2. \end{aligned}$$

(74)

In the third term on the left-hand side one recognizes the familiar influence of the field strength on the spin tensor, where $f_{kl}$ is the pure trace part already removed from $\varPhi _{kl}$, which can well be called field strength in the proper sense and which is, as mentioned several times, still completely undetermined by the metric.

The second term seems to me to be of considerable theoretical interest. It is, however, too small by many, many powers of ten to be able to replace, for example, the term on the right-hand side. For $\mu$ is the reciprocal Compton wavelength, about $10^{11}\mathrm{cm}^{-1}$. At least it seems significant that one naturally meets in the generalized theory a term at all similar to the enigmatic mass term (see also [9]).